LMFDB - Number field 24.0.419626218829023046958936634321984000000000000000000.3

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 4*x^23 + 73*x^22 - 64*x^21 + 2595*x^20 + 194*x^19 + 64108*x^18 + 69106*x^17 + 1048576*x^16 + 1235438*x^15 + 11305562*x^14 + 14569876*x^13 + 83892415*x^12 + 87789942*x^11 + 376397320*x^10 + 358161378*x^9 + 1087344601*x^8 + 565440970*x^7 + 843950596*x^6 + 157258846*x^5 + 429229107*x^4 + 99376060*x^3 + 29844986*x^2 - 533482*x + 10201)

gp: K = bnfinit(y^24 - 4*y^23 + 73*y^22 - 64*y^21 + 2595*y^20 + 194*y^19 + 64108*y^18 + 69106*y^17 + 1048576*y^16 + 1235438*y^15 + 11305562*y^14 + 14569876*y^13 + 83892415*y^12 + 87789942*y^11 + 376397320*y^10 + 358161378*y^9 + 1087344601*y^8 + 565440970*y^7 + 843950596*y^6 + 157258846*y^5 + 429229107*y^4 + 99376060*y^3 + 29844986*y^2 - 533482*y + 10201, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 4*x^23 + 73*x^22 - 64*x^21 + 2595*x^20 + 194*x^19 + 64108*x^18 + 69106*x^17 + 1048576*x^16 + 1235438*x^15 + 11305562*x^14 + 14569876*x^13 + 83892415*x^12 + 87789942*x^11 + 376397320*x^10 + 358161378*x^9 + 1087344601*x^8 + 565440970*x^7 + 843950596*x^6 + 157258846*x^5 + 429229107*x^4 + 99376060*x^3 + 29844986*x^2 - 533482*x + 10201);

oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^24 - 4*x^23 + 73*x^22 - 64*x^21 + 2595*x^20 + 194*x^19 + 64108*x^18 + 69106*x^17 + 1048576*x^16 + 1235438*x^15 + 11305562*x^14 + 14569876*x^13 + 83892415*x^12 + 87789942*x^11 + 376397320*x^10 + 358161378*x^9 + 1087344601*x^8 + 565440970*x^7 + 843950596*x^6 + 157258846*x^5 + 429229107*x^4 + 99376060*x^3 + 29844986*x^2 - 533482*x + 10201)

$ x^{24} - 4 x^{23} + 73 x^{22} - 64 x^{21} + 2595 x^{20} + 194 x^{19} + 64108 x^{18} + 69106 x^{17} + \cdots + 10201 $ x^24 - 4*x^23 + 73*x^22 - 64*x^21 + 2595*x^20 + 194*x^19 + 64108*x^18 + 69106*x^17 + 1048576*x^16 + 1235438*x^15 + 11305562*x^14 + 14569876*x^13 + 83892415*x^12 + 87789942*x^11 + 376397320*x^10 + 358161378*x^9 + 1087344601*x^8 + 565440970*x^7 + 843950596*x^6 + 157258846*x^5 + 429229107*x^4 + 99376060*x^3 + 29844986*x^2 - 533482*x + 10201

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$24$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[0, 12]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$419626218829023046958936634321984000000000000000000$ 419626218829023046958936634321984000000000000000000 $\medspace = 2^{24}\cdot 3^{12}\cdot 5^{18}\cdot 37^{16}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$128.61$	sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) magma: Abs(Discriminant(OK))^(1/Degree(K)); oscar: (1.0 * dK)^(1/degree(K))
Galois root discriminant:	$2\cdot 3^{1/2}5^{3/4}37^{2/3}\approx 128.61331797130134$
Ramified primes:	$2$, $3$, $5$, $37$ 2, 3, 5, 37	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q$
$\card{ \Gal(K/\Q) }$:	$24$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is Galois and abelian over $\Q$.
Conductor:	$2220=2^{2}\cdot 3\cdot 5\cdot 37$
Dirichlet character group:	$\lbrace$$\chi_{2220}(1,·)$, $\chi_{2220}(1987,·)$, $\chi_{2220}(581,·)$, $\chi_{2220}(1543,·)$, $\chi_{2220}(1481,·)$, $\chi_{2220}(269,·)$, $\chi_{2220}(1247,·)$, $\chi_{2220}(889,·)$, $\chi_{2220}(787,·)$, $\chi_{2220}(149,·)$, $\chi_{2220}(1601,·)$, $\chi_{2220}(343,·)$, $\chi_{2220}(667,·)$, $\chi_{2220}(1823,·)$, $\chi_{2220}(2209,·)$, $\chi_{2220}(803,·)$, $\chi_{2220}(1703,·)$, $\chi_{2220}(1321,·)$, $\chi_{2220}(47,·)$, $\chi_{2220}(1009,·)$, $\chi_{2220}(2147,·)$, $\chi_{2220}(121,·)$, $\chi_{2220}(223,·)$, $\chi_{2220}(1469,·)$$\rbrace$
This is a CM field.
Reflex fields:	unavailable$^{2048}$

Integral basis (with respect to field generator $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{3873479}a^{20}-\frac{426662}{3873479}a^{19}-\frac{171623}{3873479}a^{18}-\frac{815012}{3873479}a^{17}-\frac{274941}{3873479}a^{16}-\frac{1646692}{3873479}a^{15}+\frac{329458}{3873479}a^{14}-\frac{292666}{3873479}a^{13}-\frac{990504}{3873479}a^{12}-\frac{1721268}{3873479}a^{11}-\frac{1289875}{3873479}a^{10}-\frac{1196948}{3873479}a^{9}-\frac{1185441}{3873479}a^{8}+\frac{676018}{3873479}a^{7}+\frac{1745033}{3873479}a^{6}+\frac{484759}{3873479}a^{5}-\frac{1934343}{3873479}a^{4}-\frac{714317}{3873479}a^{3}-\frac{1734239}{3873479}a^{2}+\frac{952024}{3873479}a+\frac{954050}{3873479}$, $\frac{1}{42608269}a^{21}+\frac{3}{42608269}a^{20}-\frac{15515206}{42608269}a^{19}+\frac{9525146}{42608269}a^{18}+\frac{15827741}{42608269}a^{17}-\frac{1036942}{42608269}a^{16}+\frac{20969609}{42608269}a^{15}-\frac{648006}{42608269}a^{14}+\frac{21254003}{42608269}a^{13}-\frac{18551528}{42608269}a^{12}-\frac{13850090}{42608269}a^{11}-\frac{517362}{3873479}a^{10}-\frac{13659022}{42608269}a^{9}+\frac{9506094}{42608269}a^{8}-\frac{11395690}{42608269}a^{7}-\frac{11270197}{42608269}a^{6}+\frac{3353187}{42608269}a^{5}+\frac{5000118}{42608269}a^{4}+\frac{20518508}{42608269}a^{3}-\frac{14551894}{42608269}a^{2}-\frac{7721762}{42608269}a+\frac{3582098}{42608269}$, $\frac{1}{10\!\cdots\!09}a^{22}+\frac{483034}{10\!\cdots\!09}a^{21}-\frac{1932079}{10\!\cdots\!09}a^{20}+\frac{331980108675750}{10\!\cdots\!09}a^{19}-\frac{253122061130618}{10\!\cdots\!09}a^{18}+\frac{40962040199483}{95047663984219}a^{17}+\frac{157064864270727}{10\!\cdots\!09}a^{16}+\frac{510147534721531}{10\!\cdots\!09}a^{15}-\frac{208787344468233}{10\!\cdots\!09}a^{14}-\frac{412671397691846}{10\!\cdots\!09}a^{13}-\frac{492431761015794}{10\!\cdots\!09}a^{12}+\frac{104916675136604}{10\!\cdots\!09}a^{11}+\frac{86033227560090}{10\!\cdots\!09}a^{10}+\frac{398643121378322}{10\!\cdots\!09}a^{9}-\frac{387238828238726}{10\!\cdots\!09}a^{8}+\frac{434195094898779}{10\!\cdots\!09}a^{7}-\frac{387059375965223}{10\!\cdots\!09}a^{6}+\frac{13284394323338}{95047663984219}a^{5}-\frac{290858442627998}{10\!\cdots\!09}a^{4}+\frac{32807549363200}{10\!\cdots\!09}a^{3}+\frac{288468795213567}{10\!\cdots\!09}a^{2}+\frac{310437901499612}{10\!\cdots\!09}a-\frac{238476642762471}{10\!\cdots\!09}$, $\frac{1}{28\!\cdots\!91}a^{23}-\frac{10\!\cdots\!20}{28\!\cdots\!91}a^{22}-\frac{11\!\cdots\!22}{26\!\cdots\!81}a^{21}+\frac{91\!\cdots\!05}{26\!\cdots\!81}a^{20}+\frac{56\!\cdots\!63}{28\!\cdots\!91}a^{19}+\frac{12\!\cdots\!65}{28\!\cdots\!91}a^{18}+\frac{12\!\cdots\!33}{28\!\cdots\!91}a^{17}+\frac{48\!\cdots\!65}{28\!\cdots\!91}a^{16}+\frac{36\!\cdots\!53}{28\!\cdots\!91}a^{15}+\frac{90\!\cdots\!04}{28\!\cdots\!91}a^{14}+\frac{50\!\cdots\!42}{28\!\cdots\!91}a^{13}+\frac{12\!\cdots\!72}{28\!\cdots\!91}a^{12}-\frac{69\!\cdots\!17}{26\!\cdots\!81}a^{11}+\frac{46\!\cdots\!53}{28\!\cdots\!91}a^{10}-\frac{12\!\cdots\!86}{28\!\cdots\!91}a^{9}+\frac{19\!\cdots\!29}{28\!\cdots\!91}a^{8}+\frac{96\!\cdots\!13}{28\!\cdots\!91}a^{7}+\frac{35\!\cdots\!14}{28\!\cdots\!91}a^{6}+\frac{24\!\cdots\!17}{28\!\cdots\!91}a^{5}-\frac{84\!\cdots\!63}{28\!\cdots\!91}a^{4}+\frac{10\!\cdots\!31}{26\!\cdots\!81}a^{3}+\frac{44\!\cdots\!81}{28\!\cdots\!91}a^{2}+\frac{36\!\cdots\!50}{28\!\cdots\!91}a-\frac{10\!\cdots\!53}{28\!\cdots\!91}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^10, a^11, a^12, a^13, a^14, a^15, a^16, a^17, a^18, a^19, 1/3873479*a^20 - 426662/3873479*a^19 - 171623/3873479*a^18 - 815012/3873479*a^17 - 274941/3873479*a^16 - 1646692/3873479*a^15 + 329458/3873479*a^14 - 292666/3873479*a^13 - 990504/3873479*a^12 - 1721268/3873479*a^11 - 1289875/3873479*a^10 - 1196948/3873479*a^9 - 1185441/3873479*a^8 + 676018/3873479*a^7 + 1745033/3873479*a^6 + 484759/3873479*a^5 - 1934343/3873479*a^4 - 714317/3873479*a^3 - 1734239/3873479*a^2 + 952024/3873479*a + 954050/3873479, 1/42608269*a^21 + 3/42608269*a^20 - 15515206/42608269*a^19 + 9525146/42608269*a^18 + 15827741/42608269*a^17 - 1036942/42608269*a^16 + 20969609/42608269*a^15 - 648006/42608269*a^14 + 21254003/42608269*a^13 - 18551528/42608269*a^12 - 13850090/42608269*a^11 - 517362/3873479*a^10 - 13659022/42608269*a^9 + 9506094/42608269*a^8 - 11395690/42608269*a^7 - 11270197/42608269*a^6 + 3353187/42608269*a^5 + 5000118/42608269*a^4 + 20518508/42608269*a^3 - 14551894/42608269*a^2 - 7721762/42608269*a + 3582098/42608269, 1/1045524303826409*a^22 + 483034/1045524303826409*a^21 - 1932079/1045524303826409*a^20 + 331980108675750/1045524303826409*a^19 - 253122061130618/1045524303826409*a^18 + 40962040199483/95047663984219*a^17 + 157064864270727/1045524303826409*a^16 + 510147534721531/1045524303826409*a^15 - 208787344468233/1045524303826409*a^14 - 412671397691846/1045524303826409*a^13 - 492431761015794/1045524303826409*a^12 + 104916675136604/1045524303826409*a^11 + 86033227560090/1045524303826409*a^10 + 398643121378322/1045524303826409*a^9 - 387238828238726/1045524303826409*a^8 + 434195094898779/1045524303826409*a^7 - 387059375965223/1045524303826409*a^6 + 13284394323338/95047663984219*a^5 - 290858442627998/1045524303826409*a^4 + 32807549363200/1045524303826409*a^3 + 288468795213567/1045524303826409*a^2 + 310437901499612/1045524303826409*a - 238476642762471/1045524303826409, 1/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891*a^23 - 108101640038157800306507048330239250462168500728472465285004174717236020/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891*a^22 - 113939958165038268199833611851013421248440741851325434741515170812546705026322/26022854529229078050045567230530851786642366064937819192940631131470736734589094550081*a^21 + 911677438618171231316207328479559752829546581229971478249109118112679041199105/26022854529229078050045567230530851786642366064937819192940631131470736734589094550081*a^20 + 56320895475219657030539495107761571587029351527391154488363586077094289559126815499563/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891*a^19 + 122927722775816057713517228700188268135038353691411951638152304730620132995308041894365/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891*a^18 + 120657436566704087883507397377129898326273706851531964740008162694510578652971347575533/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891*a^17 + 48491006407645882689404930043056663430740231081534040352765676200182198271783920027765/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891*a^16 + 3698565386536522912897162492271575535645356923384595473714615083526392915182647417253/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891*a^15 + 90057577419104638241117697395027140272713798751967496983306955307911754910344314719604/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891*a^14 + 50395448688281881080432508756002875749467556910738922277107752019258770709078819003642/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891*a^13 + 126046124415773756038254011144713988535549480332247970215600333342586920348820505104572/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891*a^12 - 6971076645341107897000609326221717605630071118159509996350224801830008949929978962017/26022854529229078050045567230530851786642366064937819192940631131470736734589094550081*a^11 + 46312731663009010167737512488001165186703468951317139589338481092935034165540303007553/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891*a^10 - 121323763458216347491968633499731727144292284546008296401925820909300160079067783886686/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891*a^9 + 19456030015403985494475152170731978938508480162121020667852986420521583030257203612629/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891*a^8 + 96064224803944718145437712925867281705826253091377666929292733892147884139307325722013/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891*a^7 + 3542324847322860849381380490305987359532116937845051759090063803844633944675196067614/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891*a^6 + 2494069285689856850900616045288038941377945523820617768665021270862376285454975780317/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891*a^5 - 84330373046204862886582931754634847503442635174315630823614725065545715166171072039763/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891*a^4 + 10787916894045697760178997682811669968803792932187508024610817498457381724064570881731/26022854529229078050045567230530851786642366064937819192940631131470736734589094550081*a^3 + 4445369821811399966404063003188978289319553661534983567952019314534520059468578967081/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891*a^2 + 36387477398456809433795313245412289298052381742560342967902957683629162909151394057350/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891*a - 1077420368212525086427497951795844080391776180614360233803483542872411532833891583153/2834172275460592658915853856790488808446198284300158525963831113328496080004752871791

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		Not computed
Index:		$1$
Inessential primes:		None

Class group and class number

$C_{190554}$, which has order $190554$ (assuming GRH)

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

oscar: class_group(K)

Relative class number: $190554$ (assuming GRH)

Unit group

sage: UK = K.unit_group()

magma: UK, fUK := UnitGroup(K);

oscar: UK, fUK = unit_group(OK)

Rank:	$11$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	\( \frac{5333711474796986527473876296936428226681363331703891345077052719802}{3011661600321359715533593936687123794730247419010256218835086739592196689} a^{23} - \frac{21249630331823656869535865652234681130204018129596252525354818571542}{3011661600321359715533593936687123794730247419010256218835086739592196689} a^{22} + \frac{388990503066773194557329682623015796703320867483885757845012402254124}{3011661600321359715533593936687123794730247419010256218835086739592196689} a^{21} - \frac{335018674142527050334797482982510437987243540301471651672409830055584}{3011661600321359715533593936687123794730247419010256218835086739592196689} a^{20} + \frac{13833372486585686700836838581706151800472084581505319586574482917573294}{3011661600321359715533593936687123794730247419010256218835086739592196689} a^{19} + \frac{114340696593885321411197182200277081662589504747749666466537216859228}{273787418211032701412144903335193072248204310819114201712280612690199699} a^{18} + \frac{341873583263865357020491128770380529465044245869151851232483126920315500}{3011661600321359715533593936687123794730247419010256218835086739592196689} a^{17} + \frac{374048477128069025523274212158608216568671089217862104687378309877446902}{3011661600321359715533593936687123794730247419010256218835086739592196689} a^{16} + \frac{5596802580811050070727804033717413430264038009577162307480598945495415928}{3011661600321359715533593936687123794730247419010256218835086739592196689} a^{15} + \frac{6676784378756137907854578769543779349820929119398800758115490488552442629}{3011661600321359715533593936687123794730247419010256218835086739592196689} a^{14} + \frac{60375061518650791123239933148883131754577844080134183203925752868458908704}{3011661600321359715533593936687123794730247419010256218835086739592196689} a^{13} + \frac{78638751216397208293750662189348665492313964289310756591664247103181969687}{3011661600321359715533593936687123794730247419010256218835086739592196689} a^{12} + \frac{448368509897513590688132492409556266051061116539336320603930986785852463178}{3011661600321359715533593936687123794730247419010256218835086739592196689} a^{11} + \frac{474971303115522137600351395631238765252511537595041742282083813341124580703}{3011661600321359715533593936687123794730247419010256218835086739592196689} a^{10} + \frac{2012632084770214137588535016325080832314870860985888742199218830951910712018}{3011661600321359715533593936687123794730247419010256218835086739592196689} a^{9} + \frac{1939891633502600968818101603495971065087492263053426789764590833204060762649}{3011661600321359715533593936687123794730247419010256218835086739592196689} a^{8} + \frac{5819270695414828138522659692503792835063437211040188166004020159494972146596}{3011661600321359715533593936687123794730247419010256218835086739592196689} a^{7} + \frac{3098543009122888404161888254023827793505376324621441856141813445787391198805}{3011661600321359715533593936687123794730247419010256218835086739592196689} a^{6} + \frac{4518781535084015025394447007431293845518925313427347209105054396236858488240}{3011661600321359715533593936687123794730247419010256218835086739592196689} a^{5} + \frac{896403247094331038765288227122512793117045903132851151750556998066954331553}{3011661600321359715533593936687123794730247419010256218835086739592196689} a^{4} + \frac{207457407809495231231177942817866494744001333530907847361973153846213438526}{273787418211032701412144903335193072248204310819114201712280612690199699} a^{3} + \frac{567878285154797037148074276763533098892716924221555954423286846049801651297}{3011661600321359715533593936687123794730247419010256218835086739592196689} a^{2} + \frac{157504172178439878738156693944165207099561400079997365342552054696503255058}{3011661600321359715533593936687123794730247419010256218835086739592196689} a + \frac{1944194576605187982023222186809405108697005668711257786399066037825230}{29818431686350096193401920165219047472576707118913427909258284550417789} \) (5333711474796986527473876296936428226681363331703891345077052719802)/(3011661600321359715533593936687123794730247419010256218835086739592196689)a^(23) - (21249630331823656869535865652234681130204018129596252525354818571542)/(3011661600321359715533593936687123794730247419010256218835086739592196689)a^(22) + (388990503066773194557329682623015796703320867483885757845012402254124)/(3011661600321359715533593936687123794730247419010256218835086739592196689)a^(21) - (335018674142527050334797482982510437987243540301471651672409830055584)/(3011661600321359715533593936687123794730247419010256218835086739592196689)a^(20) + (13833372486585686700836838581706151800472084581505319586574482917573294)/(3011661600321359715533593936687123794730247419010256218835086739592196689)a^(19) + (114340696593885321411197182200277081662589504747749666466537216859228)/(273787418211032701412144903335193072248204310819114201712280612690199699)a^(18) + (341873583263865357020491128770380529465044245869151851232483126920315500)/(3011661600321359715533593936687123794730247419010256218835086739592196689)a^(17) + (374048477128069025523274212158608216568671089217862104687378309877446902)/(3011661600321359715533593936687123794730247419010256218835086739592196689)a^(16) + (5596802580811050070727804033717413430264038009577162307480598945495415928)/(3011661600321359715533593936687123794730247419010256218835086739592196689)a^(15) + (6676784378756137907854578769543779349820929119398800758115490488552442629)/(3011661600321359715533593936687123794730247419010256218835086739592196689)a^(14) + (60375061518650791123239933148883131754577844080134183203925752868458908704)/(3011661600321359715533593936687123794730247419010256218835086739592196689)a^(13) + (78638751216397208293750662189348665492313964289310756591664247103181969687)/(3011661600321359715533593936687123794730247419010256218835086739592196689)a^(12) + (448368509897513590688132492409556266051061116539336320603930986785852463178)/(3011661600321359715533593936687123794730247419010256218835086739592196689)a^(11) + (474971303115522137600351395631238765252511537595041742282083813341124580703)/(3011661600321359715533593936687123794730247419010256218835086739592196689)a^(10) + (2012632084770214137588535016325080832314870860985888742199218830951910712018)/(3011661600321359715533593936687123794730247419010256218835086739592196689)a^(9) + (1939891633502600968818101603495971065087492263053426789764590833204060762649)/(3011661600321359715533593936687123794730247419010256218835086739592196689)a^(8) + (5819270695414828138522659692503792835063437211040188166004020159494972146596)/(3011661600321359715533593936687123794730247419010256218835086739592196689)a^(7) + (3098543009122888404161888254023827793505376324621441856141813445787391198805)/(3011661600321359715533593936687123794730247419010256218835086739592196689)a^(6) + (4518781535084015025394447007431293845518925313427347209105054396236858488240)/(3011661600321359715533593936687123794730247419010256218835086739592196689)a^(5) + (896403247094331038765288227122512793117045903132851151750556998066954331553)/(3011661600321359715533593936687123794730247419010256218835086739592196689)a^(4) + (207457407809495231231177942817866494744001333530907847361973153846213438526)/(273787418211032701412144903335193072248204310819114201712280612690199699)a^(3) + (567878285154797037148074276763533098892716924221555954423286846049801651297)/(3011661600321359715533593936687123794730247419010256218835086739592196689)a^(2) + (157504172178439878738156693944165207099561400079997365342552054696503255058)/(3011661600321359715533593936687123794730247419010256218835086739592196689)*a + (1944194576605187982023222186809405108697005668711257786399066037825230)/(29818431686350096193401920165219047472576707118913427909258284550417789) (order $6$)	sage: UK.torsion_generator() gp: K.tu[2] magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); oscar: torsion_units_generator(OK)
Fundamental units:	$\frac{68\!\cdots\!02}{27\!\cdots\!69}a^{23}-\frac{29\!\cdots\!75}{27\!\cdots\!69}a^{22}+\frac{50\!\cdots\!54}{27\!\cdots\!69}a^{21}-\frac{57\!\cdots\!81}{27\!\cdots\!69}a^{20}+\frac{17\!\cdots\!58}{27\!\cdots\!69}a^{19}-\frac{37\!\cdots\!86}{27\!\cdots\!69}a^{18}+\frac{43\!\cdots\!90}{27\!\cdots\!69}a^{17}+\frac{34\!\cdots\!27}{27\!\cdots\!69}a^{16}+\frac{70\!\cdots\!70}{27\!\cdots\!69}a^{15}+\frac{63\!\cdots\!35}{27\!\cdots\!69}a^{14}+\frac{74\!\cdots\!36}{27\!\cdots\!69}a^{13}+\frac{77\!\cdots\!35}{27\!\cdots\!69}a^{12}+\frac{54\!\cdots\!20}{27\!\cdots\!69}a^{11}+\frac{43\!\cdots\!19}{27\!\cdots\!69}a^{10}+\frac{23\!\cdots\!10}{27\!\cdots\!69}a^{9}+\frac{17\!\cdots\!84}{27\!\cdots\!69}a^{8}+\frac{67\!\cdots\!54}{27\!\cdots\!69}a^{7}+\frac{17\!\cdots\!26}{27\!\cdots\!69}a^{6}+\frac{46\!\cdots\!30}{27\!\cdots\!69}a^{5}-\frac{48\!\cdots\!34}{25\!\cdots\!79}a^{4}+\frac{25\!\cdots\!16}{27\!\cdots\!69}a^{3}-\frac{13\!\cdots\!44}{27\!\cdots\!69}a^{2}+\frac{23\!\cdots\!16}{27\!\cdots\!69}a-\frac{45\!\cdots\!79}{27\!\cdots\!69}$, $\frac{18\!\cdots\!36}{56\!\cdots\!21}a^{23}-\frac{78\!\cdots\!59}{56\!\cdots\!21}a^{22}+\frac{14\!\cdots\!44}{62\!\cdots\!31}a^{21}-\frac{17\!\cdots\!93}{62\!\cdots\!31}a^{20}+\frac{52\!\cdots\!56}{62\!\cdots\!31}a^{19}-\frac{11\!\cdots\!17}{62\!\cdots\!31}a^{18}+\frac{12\!\cdots\!70}{62\!\cdots\!31}a^{17}+\frac{10\!\cdots\!91}{62\!\cdots\!31}a^{16}+\frac{20\!\cdots\!37}{62\!\cdots\!31}a^{15}+\frac{18\!\cdots\!99}{62\!\cdots\!31}a^{14}+\frac{21\!\cdots\!93}{62\!\cdots\!31}a^{13}+\frac{22\!\cdots\!05}{62\!\cdots\!31}a^{12}+\frac{16\!\cdots\!17}{62\!\cdots\!31}a^{11}+\frac{11\!\cdots\!95}{56\!\cdots\!21}a^{10}+\frac{70\!\cdots\!61}{62\!\cdots\!31}a^{9}+\frac{50\!\cdots\!34}{62\!\cdots\!31}a^{8}+\frac{19\!\cdots\!77}{62\!\cdots\!31}a^{7}+\frac{51\!\cdots\!52}{62\!\cdots\!31}a^{6}+\frac{13\!\cdots\!19}{62\!\cdots\!31}a^{5}-\frac{15\!\cdots\!46}{62\!\cdots\!31}a^{4}+\frac{75\!\cdots\!67}{62\!\cdots\!31}a^{3}-\frac{38\!\cdots\!78}{62\!\cdots\!31}a^{2}+\frac{69\!\cdots\!77}{62\!\cdots\!31}a+\frac{34\!\cdots\!02}{62\!\cdots\!31}$, $\frac{14\!\cdots\!77}{26\!\cdots\!81}a^{23}-\frac{63\!\cdots\!14}{28\!\cdots\!91}a^{22}+\frac{11\!\cdots\!67}{28\!\cdots\!91}a^{21}-\frac{10\!\cdots\!39}{28\!\cdots\!91}a^{20}+\frac{41\!\cdots\!64}{28\!\cdots\!91}a^{19}+\frac{40\!\cdots\!81}{28\!\cdots\!91}a^{18}+\frac{10\!\cdots\!85}{28\!\cdots\!91}a^{17}+\frac{11\!\cdots\!77}{28\!\cdots\!91}a^{16}+\frac{15\!\cdots\!75}{26\!\cdots\!81}a^{15}+\frac{20\!\cdots\!37}{28\!\cdots\!91}a^{14}+\frac{18\!\cdots\!89}{28\!\cdots\!91}a^{13}+\frac{23\!\cdots\!51}{28\!\cdots\!91}a^{12}+\frac{13\!\cdots\!79}{28\!\cdots\!91}a^{11}+\frac{14\!\cdots\!07}{28\!\cdots\!91}a^{10}+\frac{60\!\cdots\!74}{28\!\cdots\!91}a^{9}+\frac{58\!\cdots\!69}{28\!\cdots\!91}a^{8}+\frac{17\!\cdots\!28}{28\!\cdots\!91}a^{7}+\frac{94\!\cdots\!60}{28\!\cdots\!91}a^{6}+\frac{13\!\cdots\!46}{28\!\cdots\!91}a^{5}+\frac{28\!\cdots\!12}{28\!\cdots\!91}a^{4}+\frac{69\!\cdots\!46}{28\!\cdots\!91}a^{3}+\frac{17\!\cdots\!36}{28\!\cdots\!91}a^{2}+\frac{52\!\cdots\!35}{28\!\cdots\!91}a+\frac{59\!\cdots\!34}{28\!\cdots\!91}$, $\frac{83\!\cdots\!92}{28\!\cdots\!91}a^{23}-\frac{32\!\cdots\!22}{28\!\cdots\!91}a^{22}+\frac{60\!\cdots\!34}{28\!\cdots\!91}a^{21}-\frac{45\!\cdots\!22}{28\!\cdots\!91}a^{20}+\frac{21\!\cdots\!14}{28\!\cdots\!91}a^{19}+\frac{41\!\cdots\!63}{28\!\cdots\!91}a^{18}+\frac{53\!\cdots\!30}{28\!\cdots\!91}a^{17}+\frac{63\!\cdots\!47}{28\!\cdots\!91}a^{16}+\frac{87\!\cdots\!98}{28\!\cdots\!91}a^{15}+\frac{11\!\cdots\!44}{28\!\cdots\!91}a^{14}+\frac{94\!\cdots\!74}{28\!\cdots\!91}a^{13}+\frac{13\!\cdots\!02}{28\!\cdots\!91}a^{12}+\frac{70\!\cdots\!58}{28\!\cdots\!91}a^{11}+\frac{80\!\cdots\!98}{28\!\cdots\!91}a^{10}+\frac{31\!\cdots\!38}{28\!\cdots\!91}a^{9}+\frac{33\!\cdots\!34}{28\!\cdots\!91}a^{8}+\frac{92\!\cdots\!26}{28\!\cdots\!91}a^{7}+\frac{56\!\cdots\!60}{28\!\cdots\!91}a^{6}+\frac{72\!\cdots\!88}{28\!\cdots\!91}a^{5}+\frac{19\!\cdots\!63}{28\!\cdots\!91}a^{4}+\frac{34\!\cdots\!86}{28\!\cdots\!91}a^{3}+\frac{12\!\cdots\!42}{28\!\cdots\!91}a^{2}+\frac{22\!\cdots\!88}{28\!\cdots\!91}a-\frac{40\!\cdots\!99}{28\!\cdots\!91}$, $\frac{79\!\cdots\!48}{11\!\cdots\!69}a^{23}-\frac{34\!\cdots\!43}{11\!\cdots\!69}a^{22}+\frac{58\!\cdots\!56}{11\!\cdots\!69}a^{21}-\frac{68\!\cdots\!57}{11\!\cdots\!69}a^{20}+\frac{20\!\cdots\!66}{11\!\cdots\!69}a^{19}-\frac{46\!\cdots\!63}{11\!\cdots\!69}a^{18}+\frac{50\!\cdots\!50}{11\!\cdots\!69}a^{17}+\frac{39\!\cdots\!38}{11\!\cdots\!69}a^{16}+\frac{81\!\cdots\!90}{11\!\cdots\!69}a^{15}+\frac{72\!\cdots\!96}{11\!\cdots\!69}a^{14}+\frac{86\!\cdots\!76}{11\!\cdots\!69}a^{13}+\frac{88\!\cdots\!68}{11\!\cdots\!69}a^{12}+\frac{62\!\cdots\!42}{11\!\cdots\!69}a^{11}+\frac{49\!\cdots\!14}{11\!\cdots\!69}a^{10}+\frac{27\!\cdots\!72}{11\!\cdots\!69}a^{9}+\frac{19\!\cdots\!91}{11\!\cdots\!69}a^{8}+\frac{77\!\cdots\!34}{11\!\cdots\!69}a^{7}+\frac{19\!\cdots\!65}{11\!\cdots\!69}a^{6}+\frac{53\!\cdots\!02}{11\!\cdots\!69}a^{5}-\frac{58\!\cdots\!53}{10\!\cdots\!79}a^{4}+\frac{31\!\cdots\!94}{11\!\cdots\!69}a^{3}-\frac{15\!\cdots\!02}{11\!\cdots\!69}a^{2}+\frac{27\!\cdots\!82}{11\!\cdots\!69}a-\frac{13\!\cdots\!14}{11\!\cdots\!69}$, $\frac{83\!\cdots\!58}{28\!\cdots\!91}a^{23}-\frac{43\!\cdots\!00}{28\!\cdots\!91}a^{22}+\frac{65\!\cdots\!30}{28\!\cdots\!91}a^{21}-\frac{13\!\cdots\!61}{28\!\cdots\!91}a^{20}+\frac{22\!\cdots\!66}{28\!\cdots\!91}a^{19}-\frac{23\!\cdots\!46}{26\!\cdots\!81}a^{18}+\frac{53\!\cdots\!72}{28\!\cdots\!91}a^{17}-\frac{10\!\cdots\!54}{28\!\cdots\!91}a^{16}+\frac{81\!\cdots\!20}{28\!\cdots\!91}a^{15}-\frac{66\!\cdots\!25}{28\!\cdots\!91}a^{14}+\frac{83\!\cdots\!52}{28\!\cdots\!91}a^{13}+\frac{46\!\cdots\!86}{28\!\cdots\!91}a^{12}+\frac{57\!\cdots\!58}{28\!\cdots\!91}a^{11}-\frac{11\!\cdots\!22}{28\!\cdots\!91}a^{10}+\frac{24\!\cdots\!52}{28\!\cdots\!91}a^{9}-\frac{75\!\cdots\!71}{28\!\cdots\!91}a^{8}+\frac{63\!\cdots\!42}{28\!\cdots\!91}a^{7}-\frac{57\!\cdots\!10}{28\!\cdots\!91}a^{6}+\frac{42\!\cdots\!78}{28\!\cdots\!91}a^{5}-\frac{59\!\cdots\!40}{28\!\cdots\!91}a^{4}+\frac{44\!\cdots\!70}{28\!\cdots\!91}a^{3}-\frac{31\!\cdots\!15}{28\!\cdots\!91}a^{2}+\frac{45\!\cdots\!40}{28\!\cdots\!91}a-\frac{80\!\cdots\!72}{28\!\cdots\!91}$, $\frac{60\!\cdots\!42}{28\!\cdots\!91}a^{23}-\frac{24\!\cdots\!68}{28\!\cdots\!91}a^{22}+\frac{40\!\cdots\!50}{26\!\cdots\!81}a^{21}-\frac{37\!\cdots\!41}{28\!\cdots\!91}a^{20}+\frac{15\!\cdots\!00}{28\!\cdots\!91}a^{19}+\frac{15\!\cdots\!17}{28\!\cdots\!91}a^{18}+\frac{38\!\cdots\!58}{28\!\cdots\!91}a^{17}+\frac{42\!\cdots\!31}{28\!\cdots\!91}a^{16}+\frac{63\!\cdots\!02}{28\!\cdots\!91}a^{15}+\frac{76\!\cdots\!93}{28\!\cdots\!91}a^{14}+\frac{68\!\cdots\!22}{28\!\cdots\!91}a^{13}+\frac{89\!\cdots\!67}{28\!\cdots\!91}a^{12}+\frac{50\!\cdots\!60}{28\!\cdots\!91}a^{11}+\frac{54\!\cdots\!63}{28\!\cdots\!91}a^{10}+\frac{22\!\cdots\!16}{28\!\cdots\!91}a^{9}+\frac{22\!\cdots\!09}{28\!\cdots\!91}a^{8}+\frac{66\!\cdots\!46}{28\!\cdots\!91}a^{7}+\frac{35\!\cdots\!19}{28\!\cdots\!91}a^{6}+\frac{51\!\cdots\!00}{28\!\cdots\!91}a^{5}+\frac{10\!\cdots\!40}{28\!\cdots\!91}a^{4}+\frac{25\!\cdots\!36}{28\!\cdots\!91}a^{3}+\frac{65\!\cdots\!45}{28\!\cdots\!91}a^{2}+\frac{17\!\cdots\!46}{28\!\cdots\!91}a-\frac{28\!\cdots\!66}{25\!\cdots\!81}$, $\frac{21\!\cdots\!00}{28\!\cdots\!91}a^{23}-\frac{87\!\cdots\!52}{28\!\cdots\!91}a^{22}+\frac{15\!\cdots\!05}{28\!\cdots\!91}a^{21}-\frac{14\!\cdots\!23}{28\!\cdots\!91}a^{20}+\frac{56\!\cdots\!73}{28\!\cdots\!91}a^{19}+\frac{33\!\cdots\!99}{28\!\cdots\!91}a^{18}+\frac{13\!\cdots\!65}{28\!\cdots\!91}a^{17}+\frac{14\!\cdots\!73}{28\!\cdots\!91}a^{16}+\frac{22\!\cdots\!58}{28\!\cdots\!91}a^{15}+\frac{26\!\cdots\!08}{28\!\cdots\!91}a^{14}+\frac{24\!\cdots\!12}{28\!\cdots\!91}a^{13}+\frac{31\!\cdots\!11}{28\!\cdots\!91}a^{12}+\frac{18\!\cdots\!39}{28\!\cdots\!91}a^{11}+\frac{18\!\cdots\!25}{28\!\cdots\!91}a^{10}+\frac{81\!\cdots\!56}{28\!\cdots\!91}a^{9}+\frac{77\!\cdots\!10}{28\!\cdots\!91}a^{8}+\frac{23\!\cdots\!49}{28\!\cdots\!91}a^{7}+\frac{12\!\cdots\!97}{28\!\cdots\!91}a^{6}+\frac{18\!\cdots\!70}{28\!\cdots\!91}a^{5}+\frac{32\!\cdots\!78}{28\!\cdots\!91}a^{4}+\frac{94\!\cdots\!41}{28\!\cdots\!91}a^{3}+\frac{20\!\cdots\!99}{28\!\cdots\!91}a^{2}+\frac{65\!\cdots\!96}{28\!\cdots\!91}a-\frac{11\!\cdots\!92}{28\!\cdots\!91}$, $\frac{85\!\cdots\!96}{28\!\cdots\!91}a^{23}-\frac{33\!\cdots\!34}{28\!\cdots\!91}a^{22}+\frac{62\!\cdots\!20}{28\!\cdots\!91}a^{21}-\frac{48\!\cdots\!83}{28\!\cdots\!91}a^{20}+\frac{22\!\cdots\!84}{28\!\cdots\!91}a^{19}+\frac{37\!\cdots\!56}{28\!\cdots\!91}a^{18}+\frac{54\!\cdots\!91}{28\!\cdots\!91}a^{17}+\frac{58\!\cdots\!48}{26\!\cdots\!81}a^{16}+\frac{90\!\cdots\!51}{28\!\cdots\!91}a^{15}+\frac{10\!\cdots\!13}{26\!\cdots\!81}a^{14}+\frac{97\!\cdots\!57}{28\!\cdots\!91}a^{13}+\frac{13\!\cdots\!50}{28\!\cdots\!91}a^{12}+\frac{72\!\cdots\!80}{28\!\cdots\!91}a^{11}+\frac{81\!\cdots\!00}{28\!\cdots\!91}a^{10}+\frac{29\!\cdots\!37}{26\!\cdots\!81}a^{9}+\frac{33\!\cdots\!80}{28\!\cdots\!91}a^{8}+\frac{95\!\cdots\!35}{28\!\cdots\!91}a^{7}+\frac{56\!\cdots\!38}{28\!\cdots\!91}a^{6}+\frac{75\!\cdots\!55}{28\!\cdots\!91}a^{5}+\frac{19\!\cdots\!98}{28\!\cdots\!91}a^{4}+\frac{36\!\cdots\!75}{28\!\cdots\!91}a^{3}+\frac{10\!\cdots\!67}{28\!\cdots\!91}a^{2}+\frac{24\!\cdots\!70}{28\!\cdots\!91}a-\frac{42\!\cdots\!28}{28\!\cdots\!91}$, $\frac{21\!\cdots\!04}{28\!\cdots\!91}a^{23}-\frac{85\!\cdots\!76}{28\!\cdots\!91}a^{22}+\frac{15\!\cdots\!05}{28\!\cdots\!91}a^{21}-\frac{13\!\cdots\!58}{28\!\cdots\!91}a^{20}+\frac{55\!\cdots\!35}{28\!\cdots\!91}a^{19}+\frac{42\!\cdots\!77}{28\!\cdots\!91}a^{18}+\frac{13\!\cdots\!02}{28\!\cdots\!91}a^{17}+\frac{14\!\cdots\!69}{28\!\cdots\!91}a^{16}+\frac{22\!\cdots\!29}{28\!\cdots\!91}a^{15}+\frac{26\!\cdots\!34}{28\!\cdots\!91}a^{14}+\frac{24\!\cdots\!65}{28\!\cdots\!91}a^{13}+\frac{31\!\cdots\!84}{28\!\cdots\!91}a^{12}+\frac{17\!\cdots\!37}{28\!\cdots\!91}a^{11}+\frac{18\!\cdots\!24}{28\!\cdots\!91}a^{10}+\frac{80\!\cdots\!49}{28\!\cdots\!91}a^{9}+\frac{76\!\cdots\!98}{28\!\cdots\!91}a^{8}+\frac{23\!\cdots\!60}{28\!\cdots\!91}a^{7}+\frac{12\!\cdots\!20}{28\!\cdots\!91}a^{6}+\frac{18\!\cdots\!73}{28\!\cdots\!91}a^{5}+\frac{30\!\cdots\!41}{26\!\cdots\!81}a^{4}+\frac{91\!\cdots\!86}{28\!\cdots\!91}a^{3}+\frac{21\!\cdots\!91}{28\!\cdots\!91}a^{2}+\frac{63\!\cdots\!16}{28\!\cdots\!91}a-\frac{11\!\cdots\!81}{28\!\cdots\!91}$, $\frac{15\!\cdots\!06}{28\!\cdots\!91}a^{23}-\frac{66\!\cdots\!29}{28\!\cdots\!91}a^{22}+\frac{10\!\cdots\!61}{25\!\cdots\!81}a^{21}-\frac{13\!\cdots\!63}{28\!\cdots\!91}a^{20}+\frac{36\!\cdots\!45}{25\!\cdots\!81}a^{19}-\frac{84\!\cdots\!39}{28\!\cdots\!91}a^{18}+\frac{99\!\cdots\!11}{28\!\cdots\!91}a^{17}+\frac{78\!\cdots\!96}{28\!\cdots\!91}a^{16}+\frac{14\!\cdots\!81}{25\!\cdots\!81}a^{15}+\frac{14\!\cdots\!02}{28\!\cdots\!91}a^{14}+\frac{16\!\cdots\!48}{28\!\cdots\!91}a^{13}+\frac{17\!\cdots\!16}{28\!\cdots\!91}a^{12}+\frac{12\!\cdots\!86}{28\!\cdots\!91}a^{11}+\frac{98\!\cdots\!56}{28\!\cdots\!91}a^{10}+\frac{54\!\cdots\!83}{28\!\cdots\!91}a^{9}+\frac{38\!\cdots\!33}{28\!\cdots\!91}a^{8}+\frac{15\!\cdots\!88}{28\!\cdots\!91}a^{7}+\frac{36\!\cdots\!34}{25\!\cdots\!81}a^{6}+\frac{10\!\cdots\!17}{28\!\cdots\!91}a^{5}-\frac{12\!\cdots\!49}{28\!\cdots\!91}a^{4}+\frac{59\!\cdots\!60}{28\!\cdots\!91}a^{3}-\frac{29\!\cdots\!86}{28\!\cdots\!91}a^{2}+\frac{53\!\cdots\!35}{28\!\cdots\!91}a-\frac{10\!\cdots\!19}{28\!\cdots\!91}$ 68223486172145789703775138732923982802/279791970767234834112416912376973600960535869a^23 - 292524431203481110992614541632738857075/279791970767234834112416912376973600960535869a^22 + 5059014694766338212808008145010030366054/279791970767234834112416912376973600960535869a^21 - 5799484238525775271055527043181458950281/279791970767234834112416912376973600960535869a^20 + 178306117580428305542471488444085004454558/279791970767234834112416912376973600960535869a^19 - 37672738395644287195361529307648607146186/279791970767234834112416912376973600960535869a^18 + 4370221661271797111434743776646494067371090/279791970767234834112416912376973600960535869a^17 + 3457709321172395303917649138779250309666127/279791970767234834112416912376973600960535869a^16 + 70190074214642773721870793288206550739819870/279791970767234834112416912376973600960535869a^15 + 63749427439396749282312194907784778703506635/279791970767234834112416912376973600960535869a^14 + 747219475738143219300548499425099383207165836/279791970767234834112416912376973600960535869a^13 + 772809580193645370701026012147198191041739935/279791970767234834112416912376973600960535869a^12 + 5439179513413466957846512396363983719553518120/279791970767234834112416912376973600960535869a^11 + 4350280376293813602704520447962283409816947219/279791970767234834112416912376973600960535869a^10 + 23968910514273453454500429528441265097453349610/279791970767234834112416912376973600960535869a^9 + 17096135542262514161872570748306234090248141284/279791970767234834112416912376973600960535869a^8 + 67195639926905466874994358928981506998870067454/279791970767234834112416912376973600960535869a^7 + 17439978029514878360136236444145330146265631426/279791970767234834112416912376973600960535869a^6 + 46565809583282404911660545119971474222656410230/279791970767234834112416912376973600960535869a^5 - 488692985118210877732064307136463614442554734/25435633706112257646583355670633963723685079a^4 + 25827300470401569140851093286784791470739179016/279791970767234834112416912376973600960535869a^3 - 1314396609177969735286024531050789756300904944/279791970767234834112416912376973600960535869a^2 + 23578883669655990418579362671154344522050916/279791970767234834112416912376973600960535869a - 452987333644610324948171075106192955129616079/279791970767234834112416912376973600960535869, 18247556433612806531590122550061243089796068641095276085310101636/56482856795676414617215644223402670660565371190544029843343515411146521a^23 - 78238520229533519571929276144865953134627282733186021761471339059/56482856795676414617215644223402670660565371190544029843343515411146521a^22 + 14884202572625348600693048571086971455135581129393137477726726771144/621311424752440560789372086457429377266219083095984328276778669522611731a^21 - 17060922345921595473915518775042465383966696899836730891717548815293/621311424752440560789372086457429377266219083095984328276778669522611731a^20 + 524597651046935882458995500373346253357136587834426784638660170468156/621311424752440560789372086457429377266219083095984328276778669522611731a^19 - 110770197504010100155010771696761730252893945715951317841660892460517/621311424752440560789372086457429377266219083095984328276778669522611731a^18 + 12857736162120432416185104763127996278785746834228571442098434112793070/621311424752440560789372086457429377266219083095984328276778669522611731a^17 + 10174730464183168304721594261010434432994672053102256945391182940527791/621311424752440560789372086457429377266219083095984328276778669522611731a^16 + 206509051017641145133993011493833331466105180012105519153529180779260237/621311424752440560789372086457429377266219083095984328276778669522611731a^15 + 187585438879888202448876413635132551793094279621668689218834908635861999/621311424752440560789372086457429377266219083095984328276778669522611731a^14 + 2198433094074899750069206177090184110796460163563928893759810823981075693/621311424752440560789372086457429377266219083095984328276778669522611731a^13 + 2273988985109042153233299652842791465876431961176591044012596950731366005/621311424752440560789372086457429377266219083095984328276778669522611731a^12 + 16002977942946162548025180336909414744877719618240217620846034701710031917/621311424752440560789372086457429377266219083095984328276778669522611731a^11 + 1163723306548643644663572232010465937800989424216489968807175317219895995/56482856795676414617215644223402670660565371190544029843343515411146521a^10 + 70520808357894432953301553144453267761888770884771329390804480788470677861/621311424752440560789372086457429377266219083095984328276778669522611731a^9 + 50306715442158334846888210874384705038677745958763233300224690941325859834/621311424752440560789372086457429377266219083095984328276778669522611731a^8 + 197702333585812113415190890319306152955595698724807006178693677560654781177/621311424752440560789372086457429377266219083095984328276778669522611731a^7 + 51317429860754220560211035587979711282245240351162123034429888484212947152/621311424752440560789372086457429377266219083095984328276778669522611731a^6 + 137003671877180037674423654184213546352902635084152067223415853081252385819/621311424752440560789372086457429377266219083095984328276778669522611731a^5 - 15812439667133989036198270820205900849263597404251052463776602921114382046/621311424752440560789372086457429377266219083095984328276778669522611731a^4 + 75988369987420696644575062810602764414262181383348285758700351830431559667/621311424752440560789372086457429377266219083095984328276778669522611731a^3 - 3866888910965051829499587685648426454424737069479730324241972499745966078/621311424752440560789372086457429377266219083095984328276778669522611731a^2 + 69367908064903515777570035661140401923080604975582246004784181876181777/621311424752440560789372086457429377266219083095984328276778669522611731a + 345596524624703521673187567972003319286707628028342626161494721171873102/621311424752440560789372086457429377266219083095984328276778669522611731, 145680338746910931704064997821287153463730308290118312700397324451789983963615277/26022854529229078050045567230530851786642366064937819192940631131470736734589094550081a^23 - 6374961620179749156089925973729415340359576672709285691620367040891244934878245914/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^22 + 116847723099018759635145714392645283735622611874737663720927912208364410726421880067/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^21 - 100033736072340765881713176091341805609380200366469203832009311779407579496747599839/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^20 + 4156678530416452240242567896190805238705418869813992527503163785103981164721873733564/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^19 + 401040259533363089985025080675192907364042558909918892789629154973188629534076936581/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^18 + 102755400046778558851336197546854242445366982174245129462580640008623226807195932344185/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^17 + 112977738291635703519947343779426704716513258666983186007114636423149716104811794219877/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^16 + 153013510881051018477773398510786026719427077106722938764096476742887781073653034888775/26022854529229078050045567230530851786642366064937819192940631131470736734589094550081a^15 + 2016711243764784023346147634345868962821585616824927323337552387782837925921918378989237/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^14 + 18166668159142260875207480474859813166725495082907992369614655958235083254098455431139989/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^13 + 23748525353715065359547903060224266879048620753871176897145273718479205698065896901279251/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^12 + 135014656975809447594569172004361204968899637364685892573398151768486262664731451812201979/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^11 + 143679781325132508035820227287691184668429259653333640540010559579109190767503780487051407/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^10 + 606740109065351069826170221745085383870983251208413378012626088277692507163310610233883374/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^9 + 587462694188426030021555271627324361488652293353327467894279746925121568330256149099926169/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^8 + 1757179569492188936214035352839240803228465262605271740838369087533405544700876509704468328/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^7 + 945510869062838547365066308122450067974116664771407157580319590206357661602519272686203860/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^6 + 1378324457833170880154206898195585549007598494600638988414544199941085299786472964137213446/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^5 + 282638896026080079440382217186352082122610557718750848794533100710469817985570295576103412/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^4 + 697632233464904804286153857540732287211050556913756675261220433747000605824409510075058146/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^3 + 173666378547377124399902132643984613611000811020843409008145948553502907266213086890551336/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^2 + 52991273141405674472826623715371697421798103221340933860011904153140822241411847573513435/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a + 594551927409072127231632274303995453689014711857534621364187660722796136057292924234/2834172275460592658915853856790488808446198284300158525963831113328496080004752871791, 83117858250906464987347611178260933716931950628812464296608989237314312801513992/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^23 - 322583781570117096023906337394812959953421678170654928442847740397969099531390122/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^22 + 6025315743684071488246141615516737397802187235041925413035856795719037629509342634/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^21 - 4587029516263705260382014909250426171486112563454216222244937209406196538566229922/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^20 + 214859054289340678673149148101090379578687357064027293585748147520272994017503147114/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^19 + 41942359001160896049860561998291578842897131766860387713274439099282223916887923163/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^18 + 5323340963497230558633543487007886551294908446603555998079361908108218358271195890730/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^17 + 6376727692273202356552084731891536959838515183649080468947591969707784386334310938447/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^16 + 87662918865425411014705103091088545597104840862510698460520925675393217256822278455698/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^15 + 112852191391792041680422640738472181807545142201575865700973712529714449697364870884544/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^14 + 949018635123021260672416701187433674857987172493754633252451456073588215946075189388874/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^13 + 1319206025853017411580339818219047795528046847986831119113965512708594160916254510068602/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^12 + 7085755233327796377197050193766241608978939993256036024310624105911733752715143752887558/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^11 + 8084408760294067692135235574153800099652627523892278409469823124651466163809094778390298/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^10 + 31920407014207378895714741340441433271943026157390784204748266214381226844162745561330438/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^9 + 33237053905214040493584441908929321507170238895815721168802275076498765753612601882305034/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^8 + 92872552692151756625805238418513435102835676701061124112137329126950457201886660868622526/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^7 + 56725674552329205584917796112566179502141570602864318344092635062220732190535035353898460/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^6 + 72713844032592715404249074604915372673505589214712747270579255846680239349671313153005688/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^5 + 19806648134885058080925631295144891835922931122242250489244078833585694302754532685708063/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^4 + 34909266798948208515678335265649001831918466263342520846314384340569741563572399791770886/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^3 + 12306609993757818633564991647933721230690481313809515029810389913238338553965861574171042/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^2 + 2291312042533203489945328641888596547733676276326711305406733116287810264098161816009288/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a - 405398926419460163737611382744386444393917638709628112843344292031957962492882979699/2834172275460592658915853856790488808446198284300158525963831113328496080004752871791, 7908437891991522749061340550256193405546001296940763442048/11591558905591229405635625812810381135631269534036062845536790969a^23 - 34029837354742964645025613573585554641586813960258997684343/11591558905591229405635625812810381135631269534036062845536790969a^22 + 586902810885053160437146491822659193969573703738629324930656/11591558905591229405635625812810381135631269534036062845536790969a^21 - 680942924006276015773473048506396567129599367801472128803157/11591558905591229405635625812810381135631269534036062845536790969a^20 + 20675518312041503848255400839237877389378366380036317897942866/11591558905591229405635625812810381135631269534036062845536790969a^19 - 4675256702968485126860864994282770886460071121086116826255663/11591558905591229405635625812810381135631269534036062845536790969a^18 + 506533599486497869109855918120636683010903169892829366017158850/11591558905591229405635625812810381135631269534036062845536790969a^17 + 393219841705082248173503441307227821624066171507794297124551238/11591558905591229405635625812810381135631269534036062845536790969a^16 + 8127388359206996963948699711645213059216382257500064569389312590/11591558905591229405635625812810381135631269534036062845536790969a^15 + 7265757880823156286057056971790130881596267951306698965363667396/11591558905591229405635625812810381135631269534036062845536790969a^14 + 86464690654329515509397681220918436040973321358710860497151501976/11591558905591229405635625812810381135631269534036062845536790969a^13 + 88268354668344413571991680414663835416800843107794342295772455068/11591558905591229405635625812810381135631269534036062845536790969a^12 + 628795097088339691683539001685526140648417314709635687840477756942/11591558905591229405635625812810381135631269534036062845536790969a^11 + 494713389642451995939532555433900686782758308418558519922559085214/11591558905591229405635625812810381135631269534036062845536790969a^10 + 2769048338834650555517850986497292331284757244175646233251704334572/11591558905591229405635625812810381135631269534036062845536790969a^9 + 1941494284179660576729334621212871376467831991937173366082151916191/11591558905591229405635625812810381135631269534036062845536790969a^8 + 7757373051199502536364290467866078142250108369325428316201540005134/11591558905591229405635625812810381135631269534036062845536790969a^7 + 1914751264925689956521789274444985947171298004684927635398331485865/11591558905591229405635625812810381135631269534036062845536790969a^6 + 5384387321644272915275750403632303128294935383576959674113903634002/11591558905591229405635625812810381135631269534036062845536790969a^5 - 58036936850104830586737394709302074918867407761923319012583494653/1053778082326475400512329619346398285057388139457823895048799179a^4 + 3129834084156728262377887011332451205826498614148299055748320076994/11591558905591229405635625812810381135631269534036062845536790969a^3 - 153273527034035876527989154920111024673971901567019487158307254402/11591558905591229405635625812810381135631269534036062845536790969a^2 + 2749493170826919862135984604295248344157070128659504716827660582/11591558905591229405635625812810381135631269534036062845536790969a - 13791664831773978689368740587408431001899183896405055094084349814/11591558905591229405635625812810381135631269534036062845536790969, 83039667191143496334222116307877880098946667760747980276944285311435767826270058/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^23 - 438343128753554936449735497747628500712546309511621657961215412085414933720978000/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^22 + 6517239862964578887764461756768974036377841041435735736553681486231031120962927230/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^21 - 13188901432826612390284009880140343119439089414625550617059046742933511169847011361/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^20 + 224519268873844228889304858156233263103178313006491251988952882551564583518055832566/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^19 - 23766153341928766619835388741161553195557041028164309406005361103023271722073227746/26022854529229078050045567230530851786642366064937819192940631131470736734589094550081a^18 + 5382304222022576784998439808917366391400564365908762106772149609738491845859487247072/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^17 - 1063816226488216334712612439317734938399544167411209309658314796954987257077666297754/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^16 + 81695466229876071946309669986623048789481620905686322536093583677638069520998400343320/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^15 - 6661727891138021091393865253572467657306360779023648250602908854840578759070129324825/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^14 + 839639488720392782695673117659305275870274696182373830808065375241410439575733399956452/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^13 + 46811399777064153359947032362265585380377348030163673756583705491219511564070248822586/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^12 + 5764064079084811964170548792311569802857095274936007691825476135112265085861394074367058/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^11 - 1176182502979436177243094010363817739980010802394720402171390371917881044670285618924922/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^10 + 24487836899975995192499194406112820000220199932998017761013444334934107497649864546336052/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^9 - 7567528236267646103745211392569077016442509670301945947880262891160085583441736020930571/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^8 + 63694503802498035202150200552474584480287764365818707816917688083030278730636004840206442/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^7 - 57638291663037881298398314738076002330832490851744095997934839367607944485918606183828410/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^6 + 42958804926931890118596680276971712119434862042626889342524709119716370133223882760959678/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^5 - 59425932832468229018216012902391966395626509531947513314730969434939848470816691991261740/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^4 + 44017207804270543249614753910949105496067610491364354689267746120751717386693606226865370/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^3 - 31835854470445876283476813282672374697369836097102163614408653853764402623962644523178715/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^2 + 4524809963642347743157231875397909679073109633327625117847114994154766421055351581971440/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a - 802099454665854743260107699664698947082883827489338961264075184340074695118451154072/2834172275460592658915853856790488808446198284300158525963831113328496080004752871791, 6059617652173024988964961640734677189127718515734344898526391783383356812382088742/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^23 - 24106606377031819104174370756392166310799504395752905481051894744190105735856891368/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^22 + 40162115050709981678498504984115882616620211549277456568579634273415535532443265550/26022854529229078050045567230530851786642366064937819192940631131470736734589094550081a^21 - 378022364055828641304796470829795143614977749851972122050134267201349901609827234241/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^20 + 15713244535343362602490649610763643633337421547355665208020333318740145420333453686400/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^19 + 1520506930921681609269971329631256058789078870701234421656719052508552884072120762417/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^18 + 388388282223573213595381253924179029395480931218710648853992986234035743279892969225258/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^17 + 427205262864681395343391327492525764538988760813380324696147672338390781748472405134531/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^16 + 6360437129194605888805861215427678159730605560688851451821538894000737455210994969666702/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^15 + 7621788564404790880051294510697531882193214060611974830562562690283656557707722035035993/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^14 + 68627187654183217190470670690262854711659651355431303513267458972014976538812730027381022/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^13 + 89729224800663340287365613546623992528444888113835083936359908790957581914870527090631067/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^12 + 509814049693651637408021005794657790145026947857671709971812287715919497915750249349904260/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^11 + 542451447300605792519829919786879527791399457136246890157200956573635664655083155029886563/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^10 + 2288980438481993840635034062862829776277659257675842196282319420047844032357611452814227816/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^9 + 2216463298889328093160910127520569925503655476066303807034243407615256238991821688330992309/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^8 + 6620889477773357918676285567409772443315268327503960412305106351462932148616276921789871046/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^7 + 3555555198901263066214687285212393176869475832258108304528917040032731687417914876957273619/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^6 + 5145072004809705122160369276828936161869359353650324148006896259512697678395664306218707400/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^5 + 1042696453866330769016760044224110709562380446226710939832678568984234982034799849974907940/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^4 + 2590144999852451607270393921219735824930501072406408187517263570026730830469868410586303536/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^3 + 653728365584629168991459382650832638187424386912966940616555254600030129108574310544177245/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^2 + 178280357540466259325313860394024267592328117477810063889325646456674644325815372896466046/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a - 2868241675493399357603936027437047215256256322116091118528809553626368676898180960766/257652025041872059901441259708226255313290753118196229633075555757136007273159351981, 218079388628822259092807647477248217180122383334018779678253960911969333345687300/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^23 - 875509295501951805796718318516547228448278168435221373026462044047036105923069252/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^22 + 15935187148292485793199087474508619887567439837327641149835817506609040367077758005/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^21 - 14202022938940478730738621283602469763489586276909702179954245335570788345858690823/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^20 + 566316957212914998829620195246376133372871661865588463467712697067301343766922081873/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^19 + 33757411372163198258674790185102825293713421304897975776104291529471207121084566699/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^18 + 13986848011090363250579942584391570982771738496019544976526956677875796071567184958765/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^17 + 14863038840709135818397013567437487394604025532870010895671376048397816027880420338773/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^16 + 228617829188916517475847508499651368360648141288554992057227236623611423893807953187758/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^15 + 266174130335630098770721349927694362932842135767429158037904275258835052710681095548208/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^14 + 2464163163297212631266326511975829038722506558321956403712612419639872710564171301077612/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^13 + 3143193501758385167891902278088974701527813039717542447284784837803114727482387103340211/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^12 + 18275815517900034088452596291631526299873459047207781956067474309686138618934735950722639/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^11 + 18901623217301345219043290755113772519659526611226198003858588634501859209797055100974625/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^10 + 81995856584481854096285396886226771780016699299450107712716577891709992468941240391747656/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^9 + 77038282274639583192636008825565383722423838834956980043159416329730429805727117586418210/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^8 + 236798621638409012506219664500470917805360188459117844811161252822987691291975349288196649/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^7 + 120394625716277882268146072944036275452671227118822445185341567402708232044690671670374397/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^6 + 184374032568481152443135471479229596416413877929990886343895390544877819214428662415434270/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^5 + 32162977140252739535147519941523020606158290686730422887053062577898979475893781814799278/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^4 + 94067241443121411650808992758667411733962353237774895218668256042920390215037894819518041/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^3 + 20970649254739118844962271304598637029331579587154546109441360774086526422655563795030299/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^2 + 6585251401250276577648236456873424766714634997241135952250620593512775493493596781700996/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a - 1165504778716343006838852626907128979972291193166975194291304162771810301498491901192/2834172275460592658915853856790488808446198284300158525963831113328496080004752871791, 854425673399016827503251490243800014307703411649709138922199980372706193395205896/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^23 - 3335487527159744585232415492736687628617117849360518957491097740114416797992064734/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^22 + 62035941668378233262866494736129084133314443870575662341866918912155263936965583620/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^21 - 48651456911618667379392574848739737555719781445397165012025458458543108550202518483/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^20 + 2211373500633718529691118638333727736585438539352997267073935699715672614215234740984/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^19 + 379441382262680219063016376240415326369064870059357516775592647985615195948629255556/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^18 + 54769671726385623247101807468829331062199469919315870537741125214432535312810846096891/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^17 + 5846114719321804541152846079483608511231585395438425375683412708612284578606757516548/26022854529229078050045567230530851786642366064937819192940631131470736734589094550081a^16 + 901061853150615543523228369718248503565091973189118976221393980531904806544739510939651/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^15 + 103729012038881239522146720065182400802894077295091422242495281533553936326653451391313/26022854529229078050045567230530851786642366064937819192940631131470736734589094550081a^14 + 9751973995029764957545395447451388014133723889064821371527850256928138170073086904345757/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^13 + 13364142275583995445280880967147593849102667521949503879634503038795527021956156592926350/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^12 + 72772342621939416103767276249006197857007326474888584286650176074049540231453752139255080/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^11 + 81738333245763170789577963133362461671473027920078621961738375380323131612008820331874400/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^10 + 29817712522170555679670655915370728768183631332441490879794933743767760354550073109238137/26022854529229078050045567230530851786642366064937819192940631131470736734589094550081a^9 + 335829087867960129706409062456757925399057627036652418264038748918848572356161358302842980/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^8 + 954450939787838896289495825610657595253220526292478487573933069153687113271031111154848735/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^7 + 567417673862423497627385141645415137728661893367196119103193118605377724853853189244427638/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^6 + 754492350782648403777733722274634705195278988991249851218407537047313566379777679202898955/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^5 + 191719235846032464621094628119139438217219713840304834207234575724797757836491684624458498/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^4 + 362186477682495328416946170817855735191666277816964621997522250806605705160883185057006675/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^3 + 108610374247218508245213073056710390074371675062961229597507639112746586987130836981685667/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^2 + 24039653529438013880534797515823400656260782207628544786319405524782876704783085244183170/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a - 4253555269064140842174411280838020257470575253104083661861830680007203319042462678328/2834172275460592658915853856790488808446198284300158525963831113328496080004752871791, 21425167783915844385577391306185569031304404596113868528387298643626309557401292804/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^23 - 85660249255612013131218205503724559391762013670090454189106123658502280696729587076/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^22 + 1563866844774805810454200706757067974390424539253073041764512562076309421516647663705/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^21 - 1368225710707603780764984559822011305516843397013279531631387654810989597757173177258/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^20 + 55595088508317435504392933025079363299872284803620052731641775816733477936062144033235/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^19 + 4261892926925570202633999493615065913914055739554734304883845855917305270065135223177/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^18 + 1373509838560079489590865042795995647981235880182147323673933739254098858864365193737302/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^17 + 1483195700662865837602556003685428731481449999417650578094669305641431271845582104659469/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^16 + 22468148290398537546326624603625213850805518220058950531567241058335415205099323205010429/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^15 + 26511206812066248112702135801851434261606052151738754236600619952327330473257399224068634/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^14 + 242264243975844592444179684070428437081145734548880586959052732237963938369239070663199565/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^13 + 312607470509850977174461076608959863572941350651207220102325366414103057191799964221951784/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^12 + 1797897423191435305828393664361953265699605897850999361128985368292252413867371671262061637/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^11 + 1884168939663209330861085553640146563616620753171476231177518662512410309075860625112435024/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^10 + 8067169770023825962772822796529805098677742314731296282228365049347221426645947583088841649/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^9 + 7688038280416832218568148395332820862446945247015187474597331061365594317395048588794992698/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^8 + 23307572438679068938877160846327072835092485925087232844284241102723323585123012314979882360/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^7 + 12155093929510320235109389496078716167953952216333488259706428643458741149481241120348976820/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^6 + 18094423916138688617579413449598969873628889341930835485952363035793207875334930541915644573/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^5 + 308828359538140060794785019881559031322395673547832200047791018285046586893457056458638841/26022854529229078050045567230530851786642366064937819192940631131470736734589094550081a^4 + 9193484461396892394896363523968441539460485124371115436243491296326436886075594327611458986/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^3 + 2143446696080858396499006941395317700155865692755759569435699188114731473068584901421509391/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a^2 + 638678609537468839776723551096426276025990329498495090359591187266520161212062069554959216/286251399821519858550501239535839369653066026714316011122346942446178104080480040050891a - 113033582163134497090831259400493823446208346181532198925320336634333106101096586665681/2834172275460592658915853856790488808446198284300158525963831113328496080004752871791, 15460254712562843916309838521756280020660178452896364321549340707551810693736406/2834172275460592658915853856790488808446198284300158525963831113328496080004752871791a^23 - 66242466561496787570705827204901129283909269648909907217802907713751923630855029/2834172275460592658915853856790488808446198284300158525963831113328496080004752871791a^22 + 104198017913909884641527328825831465692983850165728306801298837541921431427594761/257652025041872059901441259708226255313290753118196229633075555757136007273159351981a^21 - 1310569821568572728243121185098663507677990644747451789387649179525134501579712663/2834172275460592658915853856790488808446198284300158525963831113328496080004752871791a^20 + 3672614839958582727593608335882844625681307628902858051157712622626670884683316945/257652025041872059901441259708226255313290753118196229633075555757136007273159351981a^19 - 8414289081530130564577868894618320448958821075488425284277028334730033790094723339/2834172275460592658915853856790488808446198284300158525963831113328496080004752871791a^18 + 990196424679562648612138983935334650841371051853055939208457753266979412376199636811/2834172275460592658915853856790488808446198284300158525963831113328496080004752871791a^17 + 786452377682602551680824653424851680543024425308525257228552242174431184008432038796/2834172275460592658915853856790488808446198284300158525963831113328496080004752871791a^16 + 1445940668700485571152740550075076591389409021795649301808792300409881250409169407881/257652025041872059901441259708226255313290753118196229633075555757136007273159351981a^15 + 14488734651975322064762286668027904397207094101779215874874843703139977668826850022202/2834172275460592658915853856790488808446198284300158525963831113328496080004752871791a^14 + 169325855889469583755997178059718938730884632522897600364361438193836459807584852913348/2834172275460592658915853856790488808446198284300158525963831113328496080004752871791a^13 + 175550726319077056145252436037996582573142757211454692614777358935159214816171972537016/2834172275460592658915853856790488808446198284300158525963831113328496080004752871791a^12 + 1232651029874113894882080570577002755985260997597453536472025842270176633391345454179586/2834172275460592658915853856790488808446198284300158525963831113328496080004752871791a^11 + 988581133549890262759653409495160406398992873291760691741966395306666806221586923182356/2834172275460592658915853856790488808446198284300158525963831113328496080004752871791a^10 + 5431546153597994878718119215635470709531521747441986824097513857789806658347491461893783/2834172275460592658915853856790488808446198284300158525963831113328496080004752871791a^9 + 3885217704429149800260038916850642517107094347654283295699831938846528910964913898043133/2834172275460592658915853856790488808446198284300158525963831113328496080004752871791a^8 + 15229616669723322016200880493451087211820338903047073293006064472824526747571696973291188/2834172275460592658915853856790488808446198284300158525963831113328496080004752871791a^7 + 361558532311189459567255198320341503574909700225566870339944864042580785240672341430234/257652025041872059901441259708226255313290753118196229633075555757136007273159351981a^6 + 10551141593336969302710819164839669723539865579573498469756303800259249427231277822242817/2834172275460592658915853856790488808446198284300158525963831113328496080004752871791a^5 - 1212619729998614982292342165474618210510360939660599913829108936455342936088865049750949/2834172275460592658915853856790488808446198284300158525963831113328496080004752871791a^4 + 5941192726217664447766031464011296151248099652664346357213743017199983679631592090463060/2834172275460592658915853856790488808446198284300158525963831113328496080004752871791a^3 - 297407708775904540684544350900184178533132057664340833207372867183247962272077091229486/2834172275460592658915853856790488808446198284300158525963831113328496080004752871791a^2 + 5335202514528513030139084482837679924575108748766602918879547186565966410980568330835/2834172275460592658915853856790488808446198284300158525963831113328496080004752871791*a - 108077020990583508729036836836575827606504576152176214594843675922175735531667817080519/2834172275460592658915853856790488808446198284300158525963831113328496080004752871791 (assuming GRH)	sage: UK.fundamental_units() gp: K.fu magma: [K\|fUK(g): g in Generators(UK)]; oscar: [K(fUK(a)) for a in gens(UK)]
Regulator:	$ 22287293994.214638 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 22287293994.214638 \cdot 190554}{6\cdot\sqrt{419626218829023046958936634321984000000000000000000}}\cr\approx \mathstrut & 0.130813110843853 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^24 - 4*x^23 + 73*x^22 - 64*x^21 + 2595*x^20 + 194*x^19 + 64108*x^18 + 69106*x^17 + 1048576*x^16 + 1235438*x^15 + 11305562*x^14 + 14569876*x^13 + 83892415*x^12 + 87789942*x^11 + 376397320*x^10 + 358161378*x^9 + 1087344601*x^8 + 565440970*x^7 + 843950596*x^6 + 157258846*x^5 + 429229107*x^4 + 99376060*x^3 + 29844986*x^2 - 533482*x + 10201)

DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()

hK = K.class_number(); wK = K.unit_group().torsion_generator().order();

2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))

# self-contained Pari/GP code snippet to compute the analytic class number formula

K = bnfinit(x^24 - 4*x^23 + 73*x^22 - 64*x^21 + 2595*x^20 + 194*x^19 + 64108*x^18 + 69106*x^17 + 1048576*x^16 + 1235438*x^15 + 11305562*x^14 + 14569876*x^13 + 83892415*x^12 + 87789942*x^11 + 376397320*x^10 + 358161378*x^9 + 1087344601*x^8 + 565440970*x^7 + 843950596*x^6 + 157258846*x^5 + 429229107*x^4 + 99376060*x^3 + 29844986*x^2 - 533482*x + 10201, 1);

[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]

/* self-contained Magma code snippet to compute the analytic class number formula */

Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^24 - 4*x^23 + 73*x^22 - 64*x^21 + 2595*x^20 + 194*x^19 + 64108*x^18 + 69106*x^17 + 1048576*x^16 + 1235438*x^15 + 11305562*x^14 + 14569876*x^13 + 83892415*x^12 + 87789942*x^11 + 376397320*x^10 + 358161378*x^9 + 1087344601*x^8 + 565440970*x^7 + 843950596*x^6 + 157258846*x^5 + 429229107*x^4 + 99376060*x^3 + 29844986*x^2 - 533482*x + 10201);

OK := Integers(K); DK := Discriminant(OK);

UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);

r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);

hK := #clK; wK := #TorsionSubgroup(UK);

2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));

# self-contained Oscar code snippet to compute the analytic class number formula

Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 4*x^23 + 73*x^22 - 64*x^21 + 2595*x^20 + 194*x^19 + 64108*x^18 + 69106*x^17 + 1048576*x^16 + 1235438*x^15 + 11305562*x^14 + 14569876*x^13 + 83892415*x^12 + 87789942*x^11 + 376397320*x^10 + 358161378*x^9 + 1087344601*x^8 + 565440970*x^7 + 843950596*x^6 + 157258846*x^5 + 429229107*x^4 + 99376060*x^3 + 29844986*x^2 - 533482*x + 10201);

OK = ring_of_integers(K); DK = discriminant(OK);

UK, fUK = unit_group(OK); clK, fclK = class_group(OK);

r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);

hK = order(clK); wK = torsion_units_order(K);

2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))

Galois group

$C_2\times C_{12}$ (as 24T2):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: G = GaloisGroup(K);

oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)

An abelian group of order 24

The 24 conjugacy class representatives for $C_2\times C_{12}$

Character table for $C_2\times C_{12}$

Intermediate fields

$\Q(\sqrt{-3}) $, $\Q(\sqrt{5}) $, $\Q(\sqrt{-15}) $, 3.3.1369.1, $\Q(\sqrt{-3}, \sqrt{5})$, $\Q(\zeta_{20})^+$, 4.0.18000.1, 6.0.50602347.1, 6.6.234270125.1, 6.0.6325293375.1, 8.0.324000000.2, 12.0.40009336279818890625.1, 12.12.28099835631368000000000.1, 12.0.20484780175267272000000000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]

gp: L = nfsubfields(K); L[2..length(b)]

magma: L := Subfields(K); L[2..#L];

oscar: subfields(K)[2:end-1]

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	R	R	${\href{/padicField/7.12.0.1}{12} }^{2}$	${\href{/padicField/11.2.0.1}{2} }^{12}$	${\href{/padicField/13.12.0.1}{12} }^{2}$	${\href{/padicField/17.12.0.1}{12} }^{2}$	${\href{/padicField/19.3.0.1}{3} }^{8}$	${\href{/padicField/23.4.0.1}{4} }^{6}$	${\href{/padicField/29.2.0.1}{2} }^{12}$	${\href{/padicField/31.2.0.1}{2} }^{12}$	R	${\href{/padicField/41.6.0.1}{6} }^{4}$	${\href{/padicField/43.4.0.1}{4} }^{6}$	${\href{/padicField/47.4.0.1}{4} }^{6}$	${\href{/padicField/53.12.0.1}{12} }^{2}$	${\href{/padicField/59.6.0.1}{6} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:

p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:

p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:

p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:

p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$2$ 2	2.12.12.25	$x^{12} + 12 x^{11} + 60 x^{10} + 160 x^{9} + 308 x^{8} + 736 x^{7} + 2272 x^{6} + 5632 x^{5} + 10608 x^{4} + 15040 x^{3} + 12224 x^{2} + 3584 x + 704$	$2$	$6$	$12$	$C_{12}$	$[2]^{6}$
$2$ 2	2.12.12.25	$x^{12} + 12 x^{11} + 60 x^{10} + 160 x^{9} + 308 x^{8} + 736 x^{7} + 2272 x^{6} + 5632 x^{5} + 10608 x^{4} + 15040 x^{3} + 12224 x^{2} + 3584 x + 704$	$2$	$6$	$12$	$C_{12}$	$[2]^{6}$
$3$ 3		Deg $24$	$2$	$12$	$12$
$5$ 5		Deg $24$	$4$	$6$	$18$
$37$ 37	37.12.8.1	$x^{12} + 18 x^{10} + 220 x^{9} + 114 x^{8} + 864 x^{7} - 5754 x^{6} + 7320 x^{5} - 47346 x^{4} - 240044 x^{3} + 340080 x^{2} - 2045220 x + 8612757$	$3$	$4$	$8$	$C_{12}$	$[\ ]_{3}^{4}$
$37$ 37	37.12.8.1	$x^{12} + 18 x^{10} + 220 x^{9} + 114 x^{8} + 864 x^{7} - 5754 x^{6} + 7320 x^{5} - 47346 x^{4} - 240044 x^{3} + 340080 x^{2} - 2045220 x + 8612757$	$3$	$4$	$8$	$C_{12}$	$[\ ]_{3}^{4}$

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more