Properties

Label 18.6.225...000.1
Degree 1818
Signature [6,6][6, 6]
Discriminant 2.259×10212.259\times 10^{21}
Root discriminant 15.3615.36
Ramified primes 2,3,52,3,5
Class number 11
Class group trivial
Galois group S3×C6S_3 \times C_6 (as 18T6)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 15*x^16 - 12*x^15 - 39*x^14 + 153*x^13 - 265*x^12 + 243*x^11 + 9*x^10 - 417*x^9 + 768*x^8 - 906*x^7 + 751*x^6 - 372*x^5 + 66*x^4 + 19*x^3 - 9*x^2 + 3*x - 1)
 
gp: K = bnfinit(y^18 - 6*y^17 + 15*y^16 - 12*y^15 - 39*y^14 + 153*y^13 - 265*y^12 + 243*y^11 + 9*y^10 - 417*y^9 + 768*y^8 - 906*y^7 + 751*y^6 - 372*y^5 + 66*y^4 + 19*y^3 - 9*y^2 + 3*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 6*x^17 + 15*x^16 - 12*x^15 - 39*x^14 + 153*x^13 - 265*x^12 + 243*x^11 + 9*x^10 - 417*x^9 + 768*x^8 - 906*x^7 + 751*x^6 - 372*x^5 + 66*x^4 + 19*x^3 - 9*x^2 + 3*x - 1);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 6*x^17 + 15*x^16 - 12*x^15 - 39*x^14 + 153*x^13 - 265*x^12 + 243*x^11 + 9*x^10 - 417*x^9 + 768*x^8 - 906*x^7 + 751*x^6 - 372*x^5 + 66*x^4 + 19*x^3 - 9*x^2 + 3*x - 1)
 

x186x17+15x1612x1539x14+153x13265x12+243x11+9x10+1 x^{18} - 6 x^{17} + 15 x^{16} - 12 x^{15} - 39 x^{14} + 153 x^{13} - 265 x^{12} + 243 x^{11} + 9 x^{10} + \cdots - 1 Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  1818
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  [6,6][6, 6]
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   22594362918480000000002259436291848000000000 =21232459\medspace = 2^{12}\cdot 3^{24}\cdot 5^{9} Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  15.3615.36
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:  234/351/219.3498084783633642\cdot 3^{4/3}5^{1/2}\approx 19.349808478363364
Ramified primes:   22, 33, 55 Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  Q(5)\Q(\sqrt{5})
Aut(K/Q)\Aut(K/\Q):   C6C_6
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over Q\Q.
This is not a CM field.

Integral basis (with respect to field generator aa)

11, aa, a2a^{2}, a3a^{3}, a4a^{4}, a5a^{5}, a6a^{6}, a7a^{7}, a8a^{8}, a9a^{9}, a10a^{10}, a11a^{11}, 15a1225a1115a10+25a9+15a8+15a7+25a6+15a5+25a415a215a+15\frac{1}{5}a^{12}-\frac{2}{5}a^{11}-\frac{1}{5}a^{10}+\frac{2}{5}a^{9}+\frac{1}{5}a^{8}+\frac{1}{5}a^{7}+\frac{2}{5}a^{6}+\frac{1}{5}a^{5}+\frac{2}{5}a^{4}-\frac{1}{5}a^{2}-\frac{1}{5}a+\frac{1}{5}, 15a1325a815a715a515a415a3+25a215a+25\frac{1}{5}a^{13}-\frac{2}{5}a^{8}-\frac{1}{5}a^{7}-\frac{1}{5}a^{5}-\frac{1}{5}a^{4}-\frac{1}{5}a^{3}+\frac{2}{5}a^{2}-\frac{1}{5}a+\frac{2}{5}, 15a1425a915a815a615a515a4+25a315a2+25a\frac{1}{5}a^{14}-\frac{2}{5}a^{9}-\frac{1}{5}a^{8}-\frac{1}{5}a^{6}-\frac{1}{5}a^{5}-\frac{1}{5}a^{4}+\frac{2}{5}a^{3}-\frac{1}{5}a^{2}+\frac{2}{5}a, 15a1525a1015a915a715a615a5+25a415a3+25a2\frac{1}{5}a^{15}-\frac{2}{5}a^{10}-\frac{1}{5}a^{9}-\frac{1}{5}a^{7}-\frac{1}{5}a^{6}-\frac{1}{5}a^{5}+\frac{2}{5}a^{4}-\frac{1}{5}a^{3}+\frac{2}{5}a^{2}, 195a16295a15+295a13+795a12+3495a11+3195a10495a9+4295a8+2695a7619a61195a53395a42895a3+3395a21495a+4695\frac{1}{95}a^{16}-\frac{2}{95}a^{15}+\frac{2}{95}a^{13}+\frac{7}{95}a^{12}+\frac{34}{95}a^{11}+\frac{31}{95}a^{10}-\frac{4}{95}a^{9}+\frac{42}{95}a^{8}+\frac{26}{95}a^{7}-\frac{6}{19}a^{6}-\frac{11}{95}a^{5}-\frac{33}{95}a^{4}-\frac{28}{95}a^{3}+\frac{33}{95}a^{2}-\frac{14}{95}a+\frac{46}{95}, 11519413755a1773946431519413755a1694668211519413755a151491321761519413755a14415288871519413755a13+1072138911519413755a12+15242613383995a112853500331519413755a10782617271519413755a919149656303882751a8+2500388279969145a7707982871519413755a626278124303882751a529639928488345a4+121277203303882751a3+1973763021519413755a2+1960617271519413755a6880233461519413755\frac{1}{1519413755}a^{17}-\frac{7394643}{1519413755}a^{16}-\frac{9466821}{1519413755}a^{15}-\frac{149132176}{1519413755}a^{14}-\frac{41528887}{1519413755}a^{13}+\frac{107213891}{1519413755}a^{12}+\frac{1524261}{3383995}a^{11}-\frac{285350033}{1519413755}a^{10}-\frac{78261727}{1519413755}a^{9}-\frac{19149656}{303882751}a^{8}+\frac{25003882}{79969145}a^{7}-\frac{70798287}{1519413755}a^{6}-\frac{26278124}{303882751}a^{5}-\frac{2963992}{8488345}a^{4}+\frac{121277203}{303882751}a^{3}+\frac{197376302}{1519413755}a^{2}+\frac{196061727}{1519413755}a-\frac{688023346}{1519413755} Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Not computed
Index:  11
Inessential primes:  None

Class group and class number

Trivial group, which has order 11

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:  1111
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   1 -1  (order 22) Copy content Toggle raw display
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:   42226838488345a17234533498488345a16+107735651697669a15310143098488345a141712421248488345a13+5723255288488345a12201218718905a11+748299389351a10+2581173238488345a916217484788488345a8+26424173798488345a728939306788488345a6+21680678018488345a58587266698488345a4+95737438488345a3+933011078488345a235208038488345a+71032318488345\frac{4222683}{8488345}a^{17}-\frac{23453349}{8488345}a^{16}+\frac{10773565}{1697669}a^{15}-\frac{31014309}{8488345}a^{14}-\frac{171242124}{8488345}a^{13}+\frac{572325528}{8488345}a^{12}-\frac{2012187}{18905}a^{11}+\frac{7482993}{89351}a^{10}+\frac{258117323}{8488345}a^{9}-\frac{1621748478}{8488345}a^{8}+\frac{2642417379}{8488345}a^{7}-\frac{2893930678}{8488345}a^{6}+\frac{2168067801}{8488345}a^{5}-\frac{858726669}{8488345}a^{4}+\frac{9573743}{8488345}a^{3}+\frac{93301107}{8488345}a^{2}-\frac{3520803}{8488345}a+\frac{7103231}{8488345}, 11557952431519413755a1735247824379969145a16+158160107741519413755a15100211310091519413755a149614772265303882751a13+33284234222303882751a125935902193383995a11+2145018410881519413755a10+666117681781519413755a94700444277011519413755a8+7737249946361519413755a7171102314172303882751a6+6549979088341519413755a514714492828488345a4+99046408091519413755a3+202073045761519413755a239467379991519413755a+19557100761519413755\frac{1155795243}{1519413755}a^{17}-\frac{352478243}{79969145}a^{16}+\frac{15816010774}{1519413755}a^{15}-\frac{10021131009}{1519413755}a^{14}-\frac{9614772265}{303882751}a^{13}+\frac{33284234222}{303882751}a^{12}-\frac{593590219}{3383995}a^{11}+\frac{214501841088}{1519413755}a^{10}+\frac{66611768178}{1519413755}a^{9}-\frac{470044427701}{1519413755}a^{8}+\frac{773724994636}{1519413755}a^{7}-\frac{171102314172}{303882751}a^{6}+\frac{654997908834}{1519413755}a^{5}-\frac{1471449282}{8488345}a^{4}+\frac{9904640809}{1519413755}a^{3}+\frac{20207304576}{1519413755}a^{2}-\frac{3946737999}{1519413755}a+\frac{1955710076}{1519413755}, 5680699241519413755a1711972711979969145a16+28201165871519413755a15+45746784141519413755a14242972201481519413755a13+397723160391519413755a12501266993383995a117018081201303882751a10+20672183144303882751a91173733885391519413755a8+13733111757303882751a746640855481519413755a6810339206271519413755a5+5947238198488345a4286229311631519413755a3150481254431519413755a2+84992631591519413755a+313734811519413755\frac{568069924}{1519413755}a^{17}-\frac{119727119}{79969145}a^{16}+\frac{2820116587}{1519413755}a^{15}+\frac{4574678414}{1519413755}a^{14}-\frac{24297220148}{1519413755}a^{13}+\frac{39772316039}{1519413755}a^{12}-\frac{50126699}{3383995}a^{11}-\frac{7018081201}{303882751}a^{10}+\frac{20672183144}{303882751}a^{9}-\frac{117373388539}{1519413755}a^{8}+\frac{13733111757}{303882751}a^{7}-\frac{4664085548}{1519413755}a^{6}-\frac{81033920627}{1519413755}a^{5}+\frac{594723819}{8488345}a^{4}-\frac{28622931163}{1519413755}a^{3}-\frac{15048125443}{1519413755}a^{2}+\frac{8499263159}{1519413755}a+\frac{31373481}{1519413755}, 236667436303882751a176271069415993829a16+118291050471519413755a1518481725331519413755a14498887633471519413755a13+1333690180211519413755a123938773583383995a11+18932985192303882751a10+1312579976761519413755a93807247838771519413755a8+5150904940311519413755a75058294511471519413755a6+3139050884441519413755a53431707518488345a4175921623361519413755a364447517981519413755a2+973624470303882751a+21340595821519413755\frac{236667436}{303882751}a^{17}-\frac{62710694}{15993829}a^{16}+\frac{11829105047}{1519413755}a^{15}-\frac{1848172533}{1519413755}a^{14}-\frac{49888763347}{1519413755}a^{13}+\frac{133369018021}{1519413755}a^{12}-\frac{393877358}{3383995}a^{11}+\frac{18932985192}{303882751}a^{10}+\frac{131257997676}{1519413755}a^{9}-\frac{380724783877}{1519413755}a^{8}+\frac{515090494031}{1519413755}a^{7}-\frac{505829451147}{1519413755}a^{6}+\frac{313905088444}{1519413755}a^{5}-\frac{343170751}{8488345}a^{4}-\frac{17592162336}{1519413755}a^{3}-\frac{6444751798}{1519413755}a^{2}+\frac{973624470}{303882751}a+\frac{2134059582}{1519413755}, 68723821303882751a17375177022303882751a16+43370886241519413755a1527672602231519413755a14128757017271519413755a13+451233288491519413755a121665555373383995a11+657002774581519413755a10+1466582704303882751a91210248884541519413755a8+2191212284521519413755a72585951498331519413755a6+221310520815993829a55840028518488345a4+230574133521519413755a3+73644744431519413755a269462702181519413755a105542892303882751\frac{68723821}{303882751}a^{17}-\frac{375177022}{303882751}a^{16}+\frac{4337088624}{1519413755}a^{15}-\frac{2767260223}{1519413755}a^{14}-\frac{12875701727}{1519413755}a^{13}+\frac{45123328849}{1519413755}a^{12}-\frac{166555537}{3383995}a^{11}+\frac{65700277458}{1519413755}a^{10}+\frac{1466582704}{303882751}a^{9}-\frac{121024888454}{1519413755}a^{8}+\frac{219121228452}{1519413755}a^{7}-\frac{258595149833}{1519413755}a^{6}+\frac{2213105208}{15993829}a^{5}-\frac{584002851}{8488345}a^{4}+\frac{23057413352}{1519413755}a^{3}+\frac{7364474443}{1519413755}a^{2}-\frac{6946270218}{1519413755}a-\frac{105542892}{303882751}, 9438593921519413755a1752937882571519413755a16+121643862591519413755a1567924366881519413755a14207089100279969145a13+683401139379969145a1289893378676799a11+1529293679031519413755a10+688321497921519413755a974422776540303882751a8+117572783523303882751a76302375384231519413755a6+4625129673371519413755a59317834178488345a4120194458511519413755a3+173138500991519413755a2+12798445479969145a+150414986303882751\frac{943859392}{1519413755}a^{17}-\frac{5293788257}{1519413755}a^{16}+\frac{12164386259}{1519413755}a^{15}-\frac{6792436688}{1519413755}a^{14}-\frac{2070891002}{79969145}a^{13}+\frac{6834011393}{79969145}a^{12}-\frac{89893378}{676799}a^{11}+\frac{152929367903}{1519413755}a^{10}+\frac{68832149792}{1519413755}a^{9}-\frac{74422776540}{303882751}a^{8}+\frac{117572783523}{303882751}a^{7}-\frac{630237538423}{1519413755}a^{6}+\frac{462512967337}{1519413755}a^{5}-\frac{931783417}{8488345}a^{4}-\frac{12019445851}{1519413755}a^{3}+\frac{17313850099}{1519413755}a^{2}+\frac{127984454}{79969145}a+\frac{150414986}{303882751}, 6152672561519413755a1719382635179969145a16+1801797692303882751a1564228509471519413755a14255915431991519413755a13+935967019821519413755a123437506593383995a11+25951066393303882751a10+278970819561519413755a92633513953381519413755a8+4464249352461519413755a75011653655991519413755a6+3949390090711519413755a51875789141697669a4+110307688271519413755a3+1720674729303882751a236311408091519413755a+21026861011519413755\frac{615267256}{1519413755}a^{17}-\frac{193826351}{79969145}a^{16}+\frac{1801797692}{303882751}a^{15}-\frac{6422850947}{1519413755}a^{14}-\frac{25591543199}{1519413755}a^{13}+\frac{93596701982}{1519413755}a^{12}-\frac{343750659}{3383995}a^{11}+\frac{25951066393}{303882751}a^{10}+\frac{27897081956}{1519413755}a^{9}-\frac{263351395338}{1519413755}a^{8}+\frac{446424935246}{1519413755}a^{7}-\frac{501165365599}{1519413755}a^{6}+\frac{394939009071}{1519413755}a^{5}-\frac{187578914}{1697669}a^{4}+\frac{11030768827}{1519413755}a^{3}+\frac{1720674729}{303882751}a^{2}-\frac{3631140809}{1519413755}a+\frac{2102686101}{1519413755}, 3859373281519413755a1724142007611519413755a16+61928196841519413755a1551670933221519413755a14157194213441519413755a13+631068782331519413755a122436122313383995a11+989090822791519413755a10+7254025915993829a91757736919281519413755a8+3155806906121519413755a73643514823871519413755a6+2984238663261519413755a57825854048488345a4+172890118241519413755a3+28663628279969145a27940279491519413755a+26005662061519413755\frac{385937328}{1519413755}a^{17}-\frac{2414200761}{1519413755}a^{16}+\frac{6192819684}{1519413755}a^{15}-\frac{5167093322}{1519413755}a^{14}-\frac{15719421344}{1519413755}a^{13}+\frac{63106878233}{1519413755}a^{12}-\frac{243612231}{3383995}a^{11}+\frac{98909082279}{1519413755}a^{10}+\frac{72540259}{15993829}a^{9}-\frac{175773691928}{1519413755}a^{8}+\frac{315580690612}{1519413755}a^{7}-\frac{364351482387}{1519413755}a^{6}+\frac{298423866326}{1519413755}a^{5}-\frac{782585404}{8488345}a^{4}+\frac{17289011824}{1519413755}a^{3}+\frac{286636282}{79969145}a^{2}-\frac{794027949}{1519413755}a+\frac{2600566206}{1519413755}, 2501634921519413755a1713013801911519413755a16+26334811911519413755a157010834481519413755a142045737935303882751a13+287326380131519413755a12899394463383995a11+273625312071519413755a10+181898373921519413755a9750635578561519413755a8+23075046728303882751a71304967175141519413755a6+19700081490303882751a52449329498488345a4+202760243261519413755a366482221621519413755a212324251181519413755a3756827981519413755\frac{250163492}{1519413755}a^{17}-\frac{1301380191}{1519413755}a^{16}+\frac{2633481191}{1519413755}a^{15}-\frac{701083448}{1519413755}a^{14}-\frac{2045737935}{303882751}a^{13}+\frac{28732638013}{1519413755}a^{12}-\frac{89939446}{3383995}a^{11}+\frac{27362531207}{1519413755}a^{10}+\frac{18189837392}{1519413755}a^{9}-\frac{75063557856}{1519413755}a^{8}+\frac{23075046728}{303882751}a^{7}-\frac{130496717514}{1519413755}a^{6}+\frac{19700081490}{303882751}a^{5}-\frac{244932949}{8488345}a^{4}+\frac{20276024326}{1519413755}a^{3}-\frac{6648222162}{1519413755}a^{2}-\frac{1232425118}{1519413755}a-\frac{375682798}{1519413755}, 6774358091519413755a1738243605261519413755a16+87104748331519413755a1547315542011519413755a14148840224379969145a13+486218098779969145a123181390263383995a11+1097664410691519413755a10+432412273821519413755a92544136857121519413755a8+4092501183121519413755a74538156880711519413755a6+70023481634303882751a51650129051697669a4+5671962800303882751a3102708104031519413755a2+15530177179969145a+11309886921519413755\frac{677435809}{1519413755}a^{17}-\frac{3824360526}{1519413755}a^{16}+\frac{8710474833}{1519413755}a^{15}-\frac{4731554201}{1519413755}a^{14}-\frac{1488402243}{79969145}a^{13}+\frac{4862180987}{79969145}a^{12}-\frac{318139026}{3383995}a^{11}+\frac{109766441069}{1519413755}a^{10}+\frac{43241227382}{1519413755}a^{9}-\frac{254413685712}{1519413755}a^{8}+\frac{409250118312}{1519413755}a^{7}-\frac{453815688071}{1519413755}a^{6}+\frac{70023481634}{303882751}a^{5}-\frac{165012905}{1697669}a^{4}+\frac{5671962800}{303882751}a^{3}-\frac{10270810403}{1519413755}a^{2}+\frac{155301771}{79969145}a+\frac{1130988692}{1519413755}, 10460152091519413755a171272290328303882751a16+3116531750303882751a152211792690303882751a14446121603191519413755a13+1623754603221519413755a125917481713383995a11+43918220075303882751a10+565282515011519413755a94618720207761519413755a8+153609521877303882751a78514673451321519413755a6+131983921186303882751a577929298446755a4+35559541141519413755a3+207063626171519413755a26709401811519413755a+27698908921519413755\frac{1046015209}{1519413755}a^{17}-\frac{1272290328}{303882751}a^{16}+\frac{3116531750}{303882751}a^{15}-\frac{2211792690}{303882751}a^{14}-\frac{44612160319}{1519413755}a^{13}+\frac{162375460322}{1519413755}a^{12}-\frac{591748171}{3383995}a^{11}+\frac{43918220075}{303882751}a^{10}+\frac{56528251501}{1519413755}a^{9}-\frac{461872020776}{1519413755}a^{8}+\frac{153609521877}{303882751}a^{7}-\frac{851467345132}{1519413755}a^{6}+\frac{131983921186}{303882751}a^{5}-\frac{77929298}{446755}a^{4}+\frac{3555954114}{1519413755}a^{3}+\frac{20706362617}{1519413755}a^{2}-\frac{670940181}{1519413755}a+\frac{2769890892}{1519413755} Copy content Toggle raw display
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:  4306.402696834701 4306.402696834701
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

lims1(s1)ζK(s)=(2r1(2π)r2RhwD(26(2π)64306.402696834701122259436291848000000000(0.178379021437612 \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 4306.402696834701 \cdot 1}{2\cdot\sqrt{2259436291848000000000}}\cr\approx \mathstrut & 0.178379021437612 \end{aligned}

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 15*x^16 - 12*x^15 - 39*x^14 + 153*x^13 - 265*x^12 + 243*x^11 + 9*x^10 - 417*x^9 + 768*x^8 - 906*x^7 + 751*x^6 - 372*x^5 + 66*x^4 + 19*x^3 - 9*x^2 + 3*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 - 6*x^17 + 15*x^16 - 12*x^15 - 39*x^14 + 153*x^13 - 265*x^12 + 243*x^11 + 9*x^10 - 417*x^9 + 768*x^8 - 906*x^7 + 751*x^6 - 372*x^5 + 66*x^4 + 19*x^3 - 9*x^2 + 3*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(Rationals()); K<a> := NumberField(x^18 - 6*x^17 + 15*x^16 - 12*x^15 - 39*x^14 + 153*x^13 - 265*x^12 + 243*x^11 + 9*x^10 - 417*x^9 + 768*x^8 - 906*x^7 + 751*x^6 - 372*x^5 + 66*x^4 + 19*x^3 - 9*x^2 + 3*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 - 6*x^17 + 15*x^16 - 12*x^15 - 39*x^14 + 153*x^13 - 265*x^12 + 243*x^11 + 9*x^10 - 417*x^9 + 768*x^8 - 906*x^7 + 751*x^6 - 372*x^5 + 66*x^4 + 19*x^3 - 9*x^2 + 3*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

C6×S3C_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 S3×C6S_3 \times C_6
Character table for S3×C6S_3 \times C_6

Intermediate fields

Q(5)\Q(\sqrt{5}) , Q(ζ9)+\Q(\zeta_{9})^+, 3.1.1620.1, 6.6.820125.1, 6.2.13122000.3, 9.3.4251528000.1

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: 12.0.419904000000.1
Degree 18 sibling: 18.0.1156831381426176000000.1
Minimal sibling: 12.0.419904000000.1

Frobenius cycle types

pp 22 33 55 77 1111 1313 1717 1919 2323 2929 3131 3737 4141 4343 4747 5353 5959
Cycle type R R R 63{\href{/padicField/7.6.0.1}{6} }^{3} 62,32{\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2} 63{\href{/padicField/13.6.0.1}{6} }^{3} 29{\href{/padicField/17.2.0.1}{2} }^{9} 26,16{\href{/padicField/19.2.0.1}{2} }^{6}{,}\,{\href{/padicField/19.1.0.1}{1} }^{6} 63{\href{/padicField/23.6.0.1}{6} }^{3} 36{\href{/padicField/29.3.0.1}{3} }^{6} 62,32{\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2} 29{\href{/padicField/37.2.0.1}{2} }^{9} 36{\href{/padicField/41.3.0.1}{3} }^{6} 63{\href{/padicField/43.6.0.1}{6} }^{3} 63{\href{/padicField/47.6.0.1}{6} }^{3} 29{\href{/padicField/53.2.0.1}{2} }^{9} 62,32{\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}

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 [ei,fi][e_i,f_i] for the factorization of the ideal pOKp\mathcal{O}_K for p=7p=7 in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of [ei,fi][e_i,f_i] for the factorization of the ideal pOKp\mathcal{O}_K for p=7p=7 in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of [ei,fi][e_i,f_i] for the factorization of the ideal pOKp\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 [ei,fi][e_i,f_i] for the factorization of the ideal pOKp\mathcal{O}_K for p=7p=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

ppLabelPolynomial ee ff cc Galois group Slope content
22 Copy content Toggle raw display 2.6.1.0a1.1x6+x4+x3+x+1x^{6} + x^{4} + x^{3} + x + 1116600C6C_6[ ]6[\ ]^{6}
2.6.2.12a1.1x12+2x10+2x9+x8+4x7+5x6+2x5+6x4+4x3+x2+4x+5x^{12} + 2 x^{10} + 2 x^{9} + x^{8} + 4 x^{7} + 5 x^{6} + 2 x^{5} + 6 x^{4} + 4 x^{3} + x^{2} + 4 x + 522661212C6×C2C_6\times C_2[2]6[2]^{6}
33 Copy content Toggle raw display 3.6.3.24a2.1x18+6x16+15x14+6x13+32x12+24x11+63x10+36x9+90x8+72x7+109x6+102x5+96x4+56x3+84x2+72x+35x^{18} + 6 x^{16} + 15 x^{14} + 6 x^{13} + 32 x^{12} + 24 x^{11} + 63 x^{10} + 36 x^{9} + 90 x^{8} + 72 x^{7} + 109 x^{6} + 102 x^{5} + 96 x^{4} + 56 x^{3} + 84 x^{2} + 72 x + 3533662424not computednot computed
55 Copy content Toggle raw display 5.3.2.3a1.2x6+6x4+6x3+9x2+18x+14x^{6} + 6 x^{4} + 6 x^{3} + 9 x^{2} + 18 x + 14223333C6C_6[ ]23[\ ]_{2}^{3}
5.3.2.3a1.2x6+6x4+6x3+9x2+18x+14x^{6} + 6 x^{4} + 6 x^{3} + 9 x^{2} + 18 x + 14223333C6C_6[ ]23[\ ]_{2}^{3}
5.3.2.3a1.2x6+6x4+6x3+9x2+18x+14x^{6} + 6 x^{4} + 6 x^{3} + 9 x^{2} + 18 x + 14223333C6C_6[ ]23[\ ]_{2}^{3}

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)