LMFDB - Number field 18.6.24496374598915344039936.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 2*x^16 - 5*x^15 - 3*x^14 + 22*x^13 - 23*x^12 + 45*x^11 + 117*x^10 - 54*x^9 - 54*x^8 - 21*x^7 + 15*x^6 + 39*x^5 - 63*x^4 + 17*x^3 + 15*x^2 - 8*x + 1)

gp: K = bnfinit(y^18 + 2*y^16 - 5*y^15 - 3*y^14 + 22*y^13 - 23*y^12 + 45*y^11 + 117*y^10 - 54*y^9 - 54*y^8 - 21*y^7 + 15*y^6 + 39*y^5 - 63*y^4 + 17*y^3 + 15*y^2 - 8*y + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 + 2*x^16 - 5*x^15 - 3*x^14 + 22*x^13 - 23*x^12 + 45*x^11 + 117*x^10 - 54*x^9 - 54*x^8 - 21*x^7 + 15*x^6 + 39*x^5 - 63*x^4 + 17*x^3 + 15*x^2 - 8*x + 1);

oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 + 2*x^16 - 5*x^15 - 3*x^14 + 22*x^13 - 23*x^12 + 45*x^11 + 117*x^10 - 54*x^9 - 54*x^8 - 21*x^7 + 15*x^6 + 39*x^5 - 63*x^4 + 17*x^3 + 15*x^2 - 8*x + 1)

$ x^{18} + 2 x^{16} - 5 x^{15} - 3 x^{14} + 22 x^{13} - 23 x^{12} + 45 x^{11} + 117 x^{10} - 54 x^{9} + \cdots + 1 $ x^18 + 2*x^16 - 5*x^15 - 3*x^14 + 22*x^13 - 23*x^12 + 45*x^11 + 117*x^10 - 54*x^9 - 54*x^8 - 21*x^7 + 15*x^6 + 39*x^5 - 63*x^4 + 17*x^3 + 15*x^2 - 8*x + 1

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$18$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[6, 6]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$24496374598915344039936$ 24496374598915344039936 $\medspace = 2^{18}\cdot 3^{9}\cdot 7^{15}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$17.53$	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:	$2^{3/2}3^{1/2}7^{5/6}\approx 24.794421938893013$
Ramified primes:	$2$, $3$, $7$ 2, 3, 7	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{21}) $
$\card{ \Aut(K/\Q) }$:	$6$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{9}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{6}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{7}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{7}-\frac{1}{3}a^{5}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{6}-\frac{1}{3}a^{4}-\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{15}a^{15}+\frac{1}{15}a^{12}+\frac{2}{15}a^{11}-\frac{1}{15}a^{9}+\frac{2}{15}a^{8}-\frac{1}{15}a^{7}-\frac{4}{15}a^{6}+\frac{1}{3}a^{5}+\frac{2}{5}a^{4}+\frac{1}{15}a^{3}-\frac{1}{5}a^{2}+\frac{2}{5}a-\frac{1}{15}$, $\frac{1}{45}a^{16}+\frac{1}{45}a^{15}+\frac{1}{9}a^{14}-\frac{4}{45}a^{13}+\frac{1}{15}a^{12}+\frac{2}{45}a^{11}-\frac{2}{15}a^{10}+\frac{1}{45}a^{9}+\frac{1}{45}a^{8}+\frac{1}{3}a^{7}-\frac{4}{45}a^{6}+\frac{11}{45}a^{5}-\frac{2}{5}a^{4}-\frac{17}{45}a^{3}-\frac{2}{45}a^{2}+\frac{4}{9}a+\frac{4}{45}$, $\frac{1}{2115940095}a^{17}-\frac{5925712}{2115940095}a^{16}+\frac{4092269}{235104455}a^{15}+\frac{4842356}{51608295}a^{14}-\frac{47617868}{423188019}a^{13}-\frac{44599873}{2115940095}a^{12}+\frac{288638171}{2115940095}a^{11}-\frac{70675631}{2115940095}a^{10}+\frac{348914594}{2115940095}a^{9}+\frac{10156910}{423188019}a^{8}-\frac{397719853}{2115940095}a^{7}+\frac{294451567}{2115940095}a^{6}+\frac{193197044}{2115940095}a^{5}+\frac{757031191}{2115940095}a^{4}-\frac{501480082}{2115940095}a^{3}+\frac{137104358}{705313365}a^{2}-\frac{19550264}{705313365}a-\frac{960084101}{2115940095}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, 1/3*a^8 - 1/3*a^7 - 1/3*a^6 + 1/3*a^4 - 1/3*a^3 + 1/3*a + 1/3, 1/3*a^9 + 1/3*a^7 - 1/3*a^6 + 1/3*a^5 - 1/3*a^3 + 1/3*a^2 - 1/3*a + 1/3, 1/3*a^10 - 1/3*a^6 + 1/3*a^4 - 1/3*a^3 - 1/3*a^2 - 1/3, 1/3*a^11 - 1/3*a^7 + 1/3*a^5 - 1/3*a^4 - 1/3*a^3 - 1/3*a, 1/3*a^12 - 1/3*a^7 - 1/3*a^5 - 1/3*a^3 - 1/3*a^2 + 1/3*a + 1/3, 1/3*a^13 - 1/3*a^7 + 1/3*a^6 + 1/3*a^3 + 1/3*a^2 - 1/3*a + 1/3, 1/3*a^14 - 1/3*a^6 - 1/3*a^4 - 1/3*a^2 - 1/3*a + 1/3, 1/15*a^15 + 1/15*a^12 + 2/15*a^11 - 1/15*a^9 + 2/15*a^8 - 1/15*a^7 - 4/15*a^6 + 1/3*a^5 + 2/5*a^4 + 1/15*a^3 - 1/5*a^2 + 2/5*a - 1/15, 1/45*a^16 + 1/45*a^15 + 1/9*a^14 - 4/45*a^13 + 1/15*a^12 + 2/45*a^11 - 2/15*a^10 + 1/45*a^9 + 1/45*a^8 + 1/3*a^7 - 4/45*a^6 + 11/45*a^5 - 2/5*a^4 - 17/45*a^3 - 2/45*a^2 + 4/9*a + 4/45, 1/2115940095*a^17 - 5925712/2115940095*a^16 + 4092269/235104455*a^15 + 4842356/51608295*a^14 - 47617868/423188019*a^13 - 44599873/2115940095*a^12 + 288638171/2115940095*a^11 - 70675631/2115940095*a^10 + 348914594/2115940095*a^9 + 10156910/423188019*a^8 - 397719853/2115940095*a^7 + 294451567/2115940095*a^6 + 193197044/2115940095*a^5 + 757031191/2115940095*a^4 - 501480082/2115940095*a^3 + 137104358/705313365*a^2 - 19550264/705313365*a - 960084101/2115940095

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

Trivial group, which has order $1$

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

oscar: class_group(K)

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:	$ -1 $ -1 (order $2$)	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:	$\frac{63744436}{51608295}a^{17}+\frac{1235372}{5734255}a^{16}+\frac{26773421}{10321659}a^{15}-\frac{99072608}{17202765}a^{14}-\frac{45684901}{10321659}a^{13}+\frac{1331593691}{51608295}a^{12}-\frac{1218088481}{51608295}a^{11}+\frac{2710585009}{51608295}a^{10}+\frac{521724959}{3440553}a^{9}-\frac{1770948052}{51608295}a^{8}-\frac{3486078202}{51608295}a^{7}-\frac{647877538}{17202765}a^{6}+\frac{652938574}{51608295}a^{5}+\frac{482062310}{10321659}a^{4}-\frac{1143869446}{17202765}a^{3}+\frac{382829902}{51608295}a^{2}+\frac{917938082}{51608295}a-\frac{54358373}{10321659}$, $\frac{8514302777}{2115940095}a^{17}+\frac{664250977}{705313365}a^{16}+\frac{17611346813}{2115940095}a^{15}-\frac{104828857}{5734255}a^{14}-\frac{6819100838}{423188019}a^{13}+\frac{35573914241}{423188019}a^{12}-\frac{152352187441}{2115940095}a^{11}+\frac{347881373138}{2115940095}a^{10}+\frac{357658464254}{705313365}a^{9}-\frac{195966848513}{2115940095}a^{8}-\frac{512606263247}{2115940095}a^{7}-\frac{6808716326}{47020891}a^{6}+\frac{57536633213}{2115940095}a^{5}+\frac{337497017093}{2115940095}a^{4}-\frac{148557897626}{705313365}a^{3}+\frac{6462709912}{423188019}a^{2}+\frac{136931556667}{2115940095}a-\frac{33450999773}{2115940095}$, $\frac{2591329}{507785}a^{17}+\frac{5858929}{4570065}a^{16}+\frac{48067759}{4570065}a^{15}-\frac{2557994}{111465}a^{14}-\frac{96324619}{4570065}a^{13}+\frac{162685784}{1523355}a^{12}-\frac{411338893}{4570065}a^{11}+\frac{104928952}{507785}a^{10}+\frac{2962723453}{4570065}a^{9}-\frac{102135565}{914013}a^{8}-\frac{94298950}{304671}a^{7}-\frac{173365376}{914013}a^{6}+\frac{26355598}{914013}a^{5}+\frac{21137151}{101557}a^{4}-\frac{1209266327}{4570065}a^{3}+\frac{81716056}{4570065}a^{2}+\frac{74578126}{914013}a-\frac{89649209}{4570065}$, $\frac{1966926967}{705313365}a^{17}+\frac{512300221}{705313365}a^{16}+\frac{4040931836}{705313365}a^{15}-\frac{215413963}{17202765}a^{14}-\frac{1648673299}{141062673}a^{13}+\frac{41091409958}{705313365}a^{12}-\frac{2275790216}{47020891}a^{11}+\frac{78915795923}{705313365}a^{10}+\frac{83568282553}{235104455}a^{9}-\frac{41119326127}{705313365}a^{8}-\frac{24645137131}{141062673}a^{7}-\frac{75567765832}{705313365}a^{6}+\frac{10255383893}{705313365}a^{5}+\frac{79558081181}{705313365}a^{4}-\frac{6777283356}{47020891}a^{3}+\frac{5539174526}{705313365}a^{2}+\frac{6619739687}{141062673}a-\frac{2478614812}{235104455}$, $\frac{4475676353}{705313365}a^{17}+\frac{457757016}{235104455}a^{16}+\frac{3136213222}{235104455}a^{15}-\frac{157916889}{5734255}a^{14}-\frac{19297810531}{705313365}a^{13}+\frac{92529491912}{705313365}a^{12}-\frac{74982444016}{705313365}a^{11}+\frac{178984076783}{705313365}a^{10}+\frac{578977338572}{705313365}a^{9}-\frac{63522136433}{705313365}a^{8}-\frac{253756852193}{705313365}a^{7}-\frac{166182774551}{705313365}a^{6}+\frac{14882138831}{705313365}a^{5}+\frac{35059490878}{141062673}a^{4}-\frac{46373674532}{141062673}a^{3}+\frac{7739926742}{705313365}a^{2}+\frac{68859701458}{705313365}a-\frac{15360324626}{705313365}$, $\frac{3004478659}{2115940095}a^{17}+\frac{308078111}{2115940095}a^{16}+\frac{393263044}{141062673}a^{15}-\frac{355855846}{51608295}a^{14}-\frac{10914006106}{2115940095}a^{13}+\frac{65304743897}{2115940095}a^{12}-\frac{61463196814}{2115940095}a^{11}+\frac{126886758937}{2115940095}a^{10}+\frac{364239705848}{2115940095}a^{9}-\frac{129146557487}{2115940095}a^{8}-\frac{198198512404}{2115940095}a^{7}-\frac{102026474336}{2115940095}a^{6}+\frac{7684484272}{423188019}a^{5}+\frac{128839115929}{2115940095}a^{4}-\frac{172837065304}{2115940095}a^{3}+\frac{11102837834}{705313365}a^{2}+\frac{5650672446}{235104455}a-\frac{3152202202}{423188019}$, $\frac{457757016}{235104455}a^{17}+\frac{91457392}{141062673}a^{16}+\frac{2954604418}{705313365}a^{15}-\frac{143189792}{17202765}a^{14}-\frac{1978462618}{235104455}a^{13}+\frac{9319370701}{235104455}a^{12}-\frac{22421359102}{705313365}a^{11}+\frac{55323205271}{705313365}a^{10}+\frac{178164386629}{705313365}a^{9}-\frac{12070329131}{705313365}a^{8}-\frac{72193571138}{705313365}a^{7}-\frac{52253006464}{705313365}a^{6}+\frac{746076623}{705313365}a^{5}+\frac{50099237579}{705313365}a^{4}-\frac{68346571259}{705313365}a^{3}+\frac{1724556163}{705313365}a^{2}+\frac{20445086198}{705313365}a-\frac{4475676353}{705313365}$, $\frac{759373754}{235104455}a^{17}+\frac{1478577623}{2115940095}a^{16}+\frac{2838464380}{423188019}a^{15}-\frac{764257414}{51608295}a^{14}-\frac{5308032946}{423188019}a^{13}+\frac{47579377952}{705313365}a^{12}-\frac{124615431056}{2115940095}a^{11}+\frac{94230216118}{705313365}a^{10}+\frac{170724236056}{423188019}a^{9}-\frac{165899380057}{2115940095}a^{8}-\frac{134476965794}{705313365}a^{7}-\frac{240188508854}{2115940095}a^{6}+\frac{56247184864}{2115940095}a^{5}+\frac{17769560839}{141062673}a^{4}-\frac{363042914218}{2115940095}a^{3}+\frac{29102556092}{2115940095}a^{2}+\frac{107009322217}{2115940095}a-\frac{5265341594}{423188019}$, $\frac{174232294}{235104455}a^{17}+\frac{223941742}{2115940095}a^{16}+\frac{3190997461}{2115940095}a^{15}-\frac{181487669}{51608295}a^{14}-\frac{5713075216}{2115940095}a^{13}+\frac{742323469}{47020891}a^{12}-\frac{31094932669}{2115940095}a^{11}+\frac{4443427990}{141062673}a^{10}+\frac{191313053707}{2115940095}a^{9}-\frac{54460922204}{2115940095}a^{8}-\frac{31312126193}{705313365}a^{7}-\frac{52274753878}{2115940095}a^{6}+\frac{15919511378}{2115940095}a^{5}+\frac{3938817389}{141062673}a^{4}-\frac{89090324039}{2115940095}a^{3}+\frac{12565026523}{2115940095}a^{2}+\frac{24188759849}{2115940095}a-\frac{8501373086}{2115940095}$, $\frac{16542466664}{2115940095}a^{17}+\frac{4609842392}{2115940095}a^{16}+\frac{34658062954}{2115940095}a^{15}-\frac{1786111966}{51608295}a^{14}-\frac{4609585619}{141062673}a^{13}+\frac{342480834163}{2115940095}a^{12}-\frac{94631736167}{705313365}a^{11}+\frac{669188185436}{2115940095}a^{10}+\frac{2114260810496}{2115940095}a^{9}-\frac{18921582419}{141062673}a^{8}-\frac{957721065422}{2115940095}a^{7}-\frac{622317132362}{2115940095}a^{6}+\frac{7738324209}{235104455}a^{5}+\frac{651301739354}{2115940095}a^{4}-\frac{848231312368}{2115940095}a^{3}+\frac{40699174696}{2115940095}a^{2}+\frac{255564984857}{2115940095}a-\frac{20245518083}{705313365}$, $\frac{9669940978}{2115940095}a^{17}+\frac{1117519036}{705313365}a^{16}+\frac{20426170933}{2115940095}a^{15}-\frac{334521904}{17202765}a^{14}-\frac{43521937631}{2115940095}a^{13}+\frac{198422713694}{2115940095}a^{12}-\frac{154362867647}{2115940095}a^{11}+\frac{380602681138}{2115940095}a^{10}+\frac{28147671841}{47020891}a^{9}-\frac{90004421659}{2115940095}a^{8}-\frac{110626081418}{423188019}a^{7}-\frac{128553006893}{705313365}a^{6}+\frac{10641263491}{2115940095}a^{5}+\frac{382456164847}{2115940095}a^{4}-\frac{160937670961}{705313365}a^{3}-\frac{20904104}{2115940095}a^{2}+\frac{29894962549}{423188019}a-\frac{28863720118}{2115940095}$ 63744436/51608295a^17 + 1235372/5734255a^16 + 26773421/10321659a^15 - 99072608/17202765a^14 - 45684901/10321659a^13 + 1331593691/51608295a^12 - 1218088481/51608295a^11 + 2710585009/51608295a^10 + 521724959/3440553a^9 - 1770948052/51608295a^8 - 3486078202/51608295a^7 - 647877538/17202765a^6 + 652938574/51608295a^5 + 482062310/10321659a^4 - 1143869446/17202765a^3 + 382829902/51608295a^2 + 917938082/51608295a - 54358373/10321659, 8514302777/2115940095a^17 + 664250977/705313365a^16 + 17611346813/2115940095a^15 - 104828857/5734255a^14 - 6819100838/423188019a^13 + 35573914241/423188019a^12 - 152352187441/2115940095a^11 + 347881373138/2115940095a^10 + 357658464254/705313365a^9 - 195966848513/2115940095a^8 - 512606263247/2115940095a^7 - 6808716326/47020891a^6 + 57536633213/2115940095a^5 + 337497017093/2115940095a^4 - 148557897626/705313365a^3 + 6462709912/423188019a^2 + 136931556667/2115940095a - 33450999773/2115940095, 2591329/507785a^17 + 5858929/4570065a^16 + 48067759/4570065a^15 - 2557994/111465a^14 - 96324619/4570065a^13 + 162685784/1523355a^12 - 411338893/4570065a^11 + 104928952/507785a^10 + 2962723453/4570065a^9 - 102135565/914013a^8 - 94298950/304671a^7 - 173365376/914013a^6 + 26355598/914013a^5 + 21137151/101557a^4 - 1209266327/4570065a^3 + 81716056/4570065a^2 + 74578126/914013a - 89649209/4570065, 1966926967/705313365a^17 + 512300221/705313365a^16 + 4040931836/705313365a^15 - 215413963/17202765a^14 - 1648673299/141062673a^13 + 41091409958/705313365a^12 - 2275790216/47020891a^11 + 78915795923/705313365a^10 + 83568282553/235104455a^9 - 41119326127/705313365a^8 - 24645137131/141062673a^7 - 75567765832/705313365a^6 + 10255383893/705313365a^5 + 79558081181/705313365a^4 - 6777283356/47020891a^3 + 5539174526/705313365a^2 + 6619739687/141062673a - 2478614812/235104455, 4475676353/705313365a^17 + 457757016/235104455a^16 + 3136213222/235104455a^15 - 157916889/5734255a^14 - 19297810531/705313365a^13 + 92529491912/705313365a^12 - 74982444016/705313365a^11 + 178984076783/705313365a^10 + 578977338572/705313365a^9 - 63522136433/705313365a^8 - 253756852193/705313365a^7 - 166182774551/705313365a^6 + 14882138831/705313365a^5 + 35059490878/141062673a^4 - 46373674532/141062673a^3 + 7739926742/705313365a^2 + 68859701458/705313365a - 15360324626/705313365, 3004478659/2115940095a^17 + 308078111/2115940095a^16 + 393263044/141062673a^15 - 355855846/51608295a^14 - 10914006106/2115940095a^13 + 65304743897/2115940095a^12 - 61463196814/2115940095a^11 + 126886758937/2115940095a^10 + 364239705848/2115940095a^9 - 129146557487/2115940095a^8 - 198198512404/2115940095a^7 - 102026474336/2115940095a^6 + 7684484272/423188019a^5 + 128839115929/2115940095a^4 - 172837065304/2115940095a^3 + 11102837834/705313365a^2 + 5650672446/235104455a - 3152202202/423188019, 457757016/235104455a^17 + 91457392/141062673a^16 + 2954604418/705313365a^15 - 143189792/17202765a^14 - 1978462618/235104455a^13 + 9319370701/235104455a^12 - 22421359102/705313365a^11 + 55323205271/705313365a^10 + 178164386629/705313365a^9 - 12070329131/705313365a^8 - 72193571138/705313365a^7 - 52253006464/705313365a^6 + 746076623/705313365a^5 + 50099237579/705313365a^4 - 68346571259/705313365a^3 + 1724556163/705313365a^2 + 20445086198/705313365a - 4475676353/705313365, 759373754/235104455a^17 + 1478577623/2115940095a^16 + 2838464380/423188019a^15 - 764257414/51608295a^14 - 5308032946/423188019a^13 + 47579377952/705313365a^12 - 124615431056/2115940095a^11 + 94230216118/705313365a^10 + 170724236056/423188019a^9 - 165899380057/2115940095a^8 - 134476965794/705313365a^7 - 240188508854/2115940095a^6 + 56247184864/2115940095a^5 + 17769560839/141062673a^4 - 363042914218/2115940095a^3 + 29102556092/2115940095a^2 + 107009322217/2115940095a - 5265341594/423188019, 174232294/235104455a^17 + 223941742/2115940095a^16 + 3190997461/2115940095a^15 - 181487669/51608295a^14 - 5713075216/2115940095a^13 + 742323469/47020891a^12 - 31094932669/2115940095a^11 + 4443427990/141062673a^10 + 191313053707/2115940095a^9 - 54460922204/2115940095a^8 - 31312126193/705313365a^7 - 52274753878/2115940095a^6 + 15919511378/2115940095a^5 + 3938817389/141062673a^4 - 89090324039/2115940095a^3 + 12565026523/2115940095a^2 + 24188759849/2115940095a - 8501373086/2115940095, 16542466664/2115940095a^17 + 4609842392/2115940095a^16 + 34658062954/2115940095a^15 - 1786111966/51608295a^14 - 4609585619/141062673a^13 + 342480834163/2115940095a^12 - 94631736167/705313365a^11 + 669188185436/2115940095a^10 + 2114260810496/2115940095a^9 - 18921582419/141062673a^8 - 957721065422/2115940095a^7 - 622317132362/2115940095a^6 + 7738324209/235104455a^5 + 651301739354/2115940095a^4 - 848231312368/2115940095a^3 + 40699174696/2115940095a^2 + 255564984857/2115940095a - 20245518083/705313365, 9669940978/2115940095a^17 + 1117519036/705313365a^16 + 20426170933/2115940095a^15 - 334521904/17202765a^14 - 43521937631/2115940095a^13 + 198422713694/2115940095a^12 - 154362867647/2115940095a^11 + 380602681138/2115940095a^10 + 28147671841/47020891a^9 - 90004421659/2115940095a^8 - 110626081418/423188019a^7 - 128553006893/705313365a^6 + 10641263491/2115940095a^5 + 382456164847/2115940095a^4 - 160937670961/705313365a^3 - 20904104/2115940095a^2 + 29894962549/423188019*a - 28863720118/2115940095	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:	$ 14420.859848954282 $	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{6}\cdot 14420.859848954282 \cdot 1}{2\cdot\sqrt{24496374598915344039936}}\cr\approx \mathstrut & 0.181413431094210 \end{aligned}\]

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

x = polygen(QQ); K.<a> = NumberField(x^18 + 2*x^16 - 5*x^15 - 3*x^14 + 22*x^13 - 23*x^12 + 45*x^11 + 117*x^10 - 54*x^9 - 54*x^8 - 21*x^7 + 15*x^6 + 39*x^5 - 63*x^4 + 17*x^3 + 15*x^2 - 8*x + 1)

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^18 + 2*x^16 - 5*x^15 - 3*x^14 + 22*x^13 - 23*x^12 + 45*x^11 + 117*x^10 - 54*x^9 - 54*x^8 - 21*x^7 + 15*x^6 + 39*x^5 - 63*x^4 + 17*x^3 + 15*x^2 - 8*x + 1, 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^18 + 2*x^16 - 5*x^15 - 3*x^14 + 22*x^13 - 23*x^12 + 45*x^11 + 117*x^10 - 54*x^9 - 54*x^8 - 21*x^7 + 15*x^6 + 39*x^5 - 63*x^4 + 17*x^3 + 15*x^2 - 8*x + 1);

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^18 + 2*x^16 - 5*x^15 - 3*x^14 + 22*x^13 - 23*x^12 + 45*x^11 + 117*x^10 - 54*x^9 - 54*x^8 - 21*x^7 + 15*x^6 + 39*x^5 - 63*x^4 + 17*x^3 + 15*x^2 - 8*x + 1);

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

$C_6\times S_3$ (as 18T6):

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)

A solvable group of order 36

The 18 conjugacy class representatives for $S_3 \times C_6$

Character table for $S_3 \times C_6$

Intermediate fields

$\Q(\sqrt{21}) $, $\Q(\zeta_{7})^+$, 3.1.1176.1, 6.2.29042496.1, $\Q(\zeta_{21})^+$, 9.3.1626379776.1

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]

Sibling fields

Galois closure:	data not computed
Degree 12 sibling:	12.0.1101670627147776.6
Degree 18 sibling:	18.0.464523844246098375868416.1
Minimal sibling:	12.0.1101670627147776.6

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	${\href{/padicField/5.3.0.1}{3} }^{6}$	R	${\href{/padicField/11.6.0.1}{6} }^{3}$	${\href{/padicField/13.2.0.1}{2} }^{9}$	${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$	${\href{/padicField/19.6.0.1}{6} }^{3}$	${\href{/padicField/23.6.0.1}{6} }^{3}$	${\href{/padicField/29.6.0.1}{6} }^{3}$	${\href{/padicField/31.6.0.1}{6} }^{3}$	${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$	${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{6}$	${\href{/padicField/43.2.0.1}{2} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }^{6}$	${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$	${\href{/padicField/53.6.0.1}{6} }^{3}$	${\href{/padicField/59.3.0.1}{3} }^{6}$

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	2.6.0.1	$x^{6} + x^{4} + x^{3} + x + 1$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
$2$ 2	2.12.18.23	$x^{12} - 12 x^{11} + 48 x^{10} - 344 x^{9} + 8244 x^{8} - 31136 x^{7} + 54848 x^{6} - 23104 x^{5} + 18864 x^{4} - 7360 x^{3} + 5120 x^{2} + 5760 x + 1472$	$2$	$6$	$18$	$C_6\times C_2$	$[3]^{6}$
$3$ 3	3.6.3.1	$x^{6} + 18 x^{2} - 27$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
	3.6.3.1	$x^{6} + 18 x^{2} - 27$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
	3.6.3.1	$x^{6} + 18 x^{2} - 27$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
$7$ 7	7.18.15.5	$x^{18} + 36 x^{17} + 540 x^{16} + 4344 x^{15} + 20160 x^{14} + 55296 x^{13} + 98757 x^{12} + 161784 x^{11} + 246024 x^{10} + 264920 x^{9} + 530640 x^{8} + 156384 x^{7} - 1885725 x^{6} - 6133212 x^{5} - 3645540 x^{4} + 5968464 x^{3} + 5011344 x^{2} + 1820448 x + 2358791$	$6$	$3$	$15$	$C_6 \times C_3$	$[\ ]_{6}^{3}$

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