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Properties

Label 36.0.59052973372...6201.1
Degree $36$
Signature $[0, 18]$
Discriminant $7^{30}\cdot 13^{30}$
Root discriminant $42.91$
Ramified primes $7, 13$
Class number $364$ (GRH)
Class group $[2, 182]$ (GRH)
Galois group $C_6^2$ (as 36T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, 15, 59, 250, 1030, 4283, -3370, 10292, -11317, 21146, -23469, 51190, -8321, 42702, 594, 33005, 1097, 28304, -9115, 16362, -6275, 8389, -2955, 3948, -820, 1190, -253, 365, -90, 110, -35, 27, -7, 6, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 + 6*x^34 - 7*x^33 + 27*x^32 - 35*x^31 + 110*x^30 - 90*x^29 + 365*x^28 - 253*x^27 + 1190*x^26 - 820*x^25 + 3948*x^24 - 2955*x^23 + 8389*x^22 - 6275*x^21 + 16362*x^20 - 9115*x^19 + 28304*x^18 + 1097*x^17 + 33005*x^16 + 594*x^15 + 42702*x^14 - 8321*x^13 + 51190*x^12 - 23469*x^11 + 21146*x^10 - 11317*x^9 + 10292*x^8 - 3370*x^7 + 4283*x^6 + 1030*x^5 + 250*x^4 + 59*x^3 + 15*x^2 + 3*x + 1)
 
gp: K = bnfinit(x^36 - x^35 + 6*x^34 - 7*x^33 + 27*x^32 - 35*x^31 + 110*x^30 - 90*x^29 + 365*x^28 - 253*x^27 + 1190*x^26 - 820*x^25 + 3948*x^24 - 2955*x^23 + 8389*x^22 - 6275*x^21 + 16362*x^20 - 9115*x^19 + 28304*x^18 + 1097*x^17 + 33005*x^16 + 594*x^15 + 42702*x^14 - 8321*x^13 + 51190*x^12 - 23469*x^11 + 21146*x^10 - 11317*x^9 + 10292*x^8 - 3370*x^7 + 4283*x^6 + 1030*x^5 + 250*x^4 + 59*x^3 + 15*x^2 + 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{36} - x^{35} + 6 x^{34} - 7 x^{33} + 27 x^{32} - 35 x^{31} + 110 x^{30} - 90 x^{29} + 365 x^{28} - 253 x^{27} + 1190 x^{26} - 820 x^{25} + 3948 x^{24} - 2955 x^{23} + 8389 x^{22} - 6275 x^{21} + 16362 x^{20} - 9115 x^{19} + 28304 x^{18} + 1097 x^{17} + 33005 x^{16} + 594 x^{15} + 42702 x^{14} - 8321 x^{13} + 51190 x^{12} - 23469 x^{11} + 21146 x^{10} - 11317 x^{9} + 10292 x^{8} - 3370 x^{7} + 4283 x^{6} + 1030 x^{5} + 250 x^{4} + 59 x^{3} + 15 x^{2} + 3 x + 1 \)

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

Invariants

Degree:  $36$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 18]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(59052973372357400276270969857784672833245876134958959716201=7^{30}\cdot 13^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.91$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(91=7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{91}(1,·)$, $\chi_{91}(3,·)$, $\chi_{91}(4,·)$, $\chi_{91}(9,·)$, $\chi_{91}(10,·)$, $\chi_{91}(12,·)$, $\chi_{91}(16,·)$, $\chi_{91}(17,·)$, $\chi_{91}(22,·)$, $\chi_{91}(23,·)$, $\chi_{91}(25,·)$, $\chi_{91}(27,·)$, $\chi_{91}(29,·)$, $\chi_{91}(30,·)$, $\chi_{91}(36,·)$, $\chi_{91}(38,·)$, $\chi_{91}(40,·)$, $\chi_{91}(43,·)$, $\chi_{91}(48,·)$, $\chi_{91}(51,·)$, $\chi_{91}(53,·)$, $\chi_{91}(55,·)$, $\chi_{91}(61,·)$, $\chi_{91}(62,·)$, $\chi_{91}(64,·)$, $\chi_{91}(66,·)$, $\chi_{91}(68,·)$, $\chi_{91}(69,·)$, $\chi_{91}(74,·)$, $\chi_{91}(75,·)$, $\chi_{91}(79,·)$, $\chi_{91}(81,·)$, $\chi_{91}(82,·)$, $\chi_{91}(87,·)$, $\chi_{91}(88,·)$, $\chi_{91}(90,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $\frac{1}{891070611726268189} a^{31} + \frac{414135339710244362}{891070611726268189} a^{30} - \frac{216547792775029726}{891070611726268189} a^{29} - \frac{385987343852553027}{891070611726268189} a^{28} + \frac{68719127092113650}{891070611726268189} a^{27} - \frac{3235385358854730}{891070611726268189} a^{26} - \frac{249167005345075668}{891070611726268189} a^{25} + \frac{267025305499179917}{891070611726268189} a^{24} + \frac{196960198876153319}{891070611726268189} a^{23} + \frac{337572080305500115}{891070611726268189} a^{22} - \frac{275524995593954760}{891070611726268189} a^{21} + \frac{111198766854825005}{891070611726268189} a^{20} + \frac{302672296476670889}{891070611726268189} a^{19} + \frac{366216229364427122}{891070611726268189} a^{18} + \frac{443668662612869859}{891070611726268189} a^{17} + \frac{159856130975663277}{891070611726268189} a^{16} + \frac{126302642719421980}{891070611726268189} a^{15} + \frac{344607775574891636}{891070611726268189} a^{14} + \frac{141553429809846308}{891070611726268189} a^{13} + \frac{162769474614137675}{891070611726268189} a^{12} + \frac{238642578377740368}{891070611726268189} a^{11} + \frac{229467026096174007}{891070611726268189} a^{10} - \frac{83895595261702555}{891070611726268189} a^{9} - \frac{171354067717107485}{891070611726268189} a^{8} - \frac{217795656779449998}{891070611726268189} a^{7} - \frac{231305644770220551}{891070611726268189} a^{6} + \frac{292667385116510676}{891070611726268189} a^{5} + \frac{79970593824781348}{891070611726268189} a^{4} + \frac{234144751300617887}{891070611726268189} a^{3} + \frac{385968731864734329}{891070611726268189} a^{2} - \frac{116101990882038413}{891070611726268189} a - \frac{388747931112795439}{891070611726268189}$, $\frac{1}{891070611726268189} a^{32} + \frac{348079524582657242}{891070611726268189} a^{30} + \frac{410907062321851088}{891070611726268189} a^{29} + \frac{438419948865166927}{891070611726268189} a^{28} + \frac{335221625895912911}{891070611726268189} a^{27} - \frac{370112003231153753}{891070611726268189} a^{26} + \frac{233957364826652115}{891070611726268189} a^{25} + \frac{140299209993018478}{891070611726268189} a^{24} + \frac{91683996091629329}{891070611726268189} a^{23} + \frac{17792707199009137}{891070611726268189} a^{22} + \frac{152566948134404711}{891070611726268189} a^{21} + \frac{171494438010473702}{891070611726268189} a^{20} - \frac{74533110244592892}{891070611726268189} a^{19} + \frac{206169093736185086}{891070611726268189} a^{18} - \frac{82406461399015537}{891070611726268189} a^{17} + \frac{306442734558098126}{891070611726268189} a^{16} - \frac{214679403552277813}{891070611726268189} a^{15} - \frac{315724174652420032}{891070611726268189} a^{14} + \frac{269496472064368543}{891070611726268189} a^{13} + \frac{363250169680403931}{891070611726268189} a^{12} - \frac{399717012384627512}{891070611726268189} a^{11} + \frac{384961811159220831}{891070611726268189} a^{10} + \frac{125469529013749294}{891070611726268189} a^{9} - \frac{412401062882592617}{891070611726268189} a^{8} - \frac{297709068712196712}{891070611726268189} a^{7} - \frac{227646566314271338}{891070611726268189} a^{6} + \frac{128501021635740675}{891070611726268189} a^{5} - \frac{153408853132411013}{891070611726268189} a^{4} + \frac{182244664319876435}{891070611726268189} a^{3} + \frac{201515294813585255}{891070611726268189} a^{2} + \frac{290422974092058101}{891070611726268189} a - \frac{28705578624189749}{891070611726268189}$, $\frac{1}{891070611726268189} a^{33} + \frac{391961034286947630}{891070611726268189} a^{30} - \frac{113550287962138560}{891070611726268189} a^{29} + \frac{291214235944340329}{891070611726268189} a^{28} - \frac{182192150333510797}{891070611726268189} a^{27} - \frac{332895031454415036}{891070611726268189} a^{26} + \frac{35948232810555190}{891070611726268189} a^{25} + \frac{401725016323998769}{891070611726268189} a^{24} - \frac{56772149296234006}{891070611726268189} a^{23} - \frac{146438032667983381}{891070611726268189} a^{22} + \frac{288139675196675220}{891070611726268189} a^{21} + \frac{356844795079739576}{891070611726268189} a^{20} + \frac{357053775677731468}{891070611726268189} a^{19} + \frac{262591747926992969}{891070611726268189} a^{18} - \frac{51829797389503875}{891070611726268189} a^{17} + \frac{1041623521255317}{2644126444291597} a^{16} + \frac{59730473363399019}{891070611726268189} a^{15} + \frac{89289562604650173}{891070611726268189} a^{14} - \frac{288412596354248329}{891070611726268189} a^{13} - \frac{230622965982126052}{891070611726268189} a^{12} - \frac{51758208407476941}{891070611726268189} a^{11} + \frac{309133239034780943}{891070611726268189} a^{10} + \frac{49156720955430865}{891070611726268189} a^{9} - \frac{407790270106632305}{891070611726268189} a^{8} - \frac{69099452118902692}{891070611726268189} a^{7} - \frac{139090342655843504}{891070611726268189} a^{6} + \frac{273817870805413169}{891070611726268189} a^{5} + \frac{292868366031717081}{891070611726268189} a^{4} + \frac{427763658632191368}{891070611726268189} a^{3} + \frac{146996345513191804}{891070611726268189} a^{2} - \frac{60257637886057751}{891070611726268189} a - \frac{211460151281603254}{891070611726268189}$, $\frac{1}{891070611726268189} a^{34} + \frac{415648882629638587}{891070611726268189} a^{30} - \frac{413652034514373093}{891070611726268189} a^{29} - \frac{181315387516238538}{891070611726268189} a^{28} - \frac{280610870202728226}{891070611726268189} a^{27} - \frac{375376058248103060}{891070611726268189} a^{26} - \frac{236666205304891190}{891070611726268189} a^{25} + \frac{206886207186274668}{891070611726268189} a^{24} + \frac{224587030989821453}{891070611726268189} a^{23} - \frac{29961581195938930}{891070611726268189} a^{22} - \frac{209396957370267984}{891070611726268189} a^{21} - \frac{152281875997420992}{891070611726268189} a^{20} + \frac{141361832568551556}{891070611726268189} a^{19} - \frac{118771999437003174}{891070611726268189} a^{18} - \frac{71807194504670874}{891070611726268189} a^{17} + \frac{111545208241193215}{891070611726268189} a^{16} - \frac{90245664524946558}{891070611726268189} a^{15} - \frac{394063061095095668}{891070611726268189} a^{14} + \frac{19071696959710740}{891070611726268189} a^{13} + \frac{233896860343204471}{891070611726268189} a^{12} - \frac{433858760446157148}{891070611726268189} a^{11} - \frac{181626821069275439}{891070611726268189} a^{10} - \frac{254095502035245780}{891070611726268189} a^{9} + \frac{113260256670390319}{891070611726268189} a^{8} - \frac{189711372516680760}{891070611726268189} a^{7} + \frac{403925165882385955}{891070611726268189} a^{6} - \frac{333715225027408270}{891070611726268189} a^{5} - \frac{283626493341792320}{891070611726268189} a^{4} - \frac{291804922704351961}{891070611726268189} a^{3} - \frac{414491813091707364}{891070611726268189} a^{2} + \frac{238802697557253839}{891070611726268189} a - \frac{371651652817833818}{891070611726268189}$, $\frac{1}{891070611726268189} a^{35} - \frac{56261615665655647}{891070611726268189} a^{28} - \frac{203524365896202640}{891070611726268189} a^{21} - \frac{407394260656743131}{891070611726268189} a^{14} - \frac{178798326490055451}{891070611726268189} a^{7} - \frac{393550456008817635}{891070611726268189}$

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

Class group and class number

$C_{2}\times C_{182}$, which has order $364$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $17$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{12486442664901100}{891070611726268189} a^{35} - \frac{31216106662252750}{891070611726268189} a^{34} + \frac{84283487988082425}{891070611726268189} a^{33} - \frac{187296639973516500}{891070611726268189} a^{32} + \frac{408943283688848980}{891070611726268189} a^{31} - \frac{858442933211950625}{891070611726268189} a^{30} + \frac{1754345194418604550}{891070611726268189} a^{29} - \frac{2771990271608044200}{891070611726268189} a^{28} + \frac{5103833439278324625}{891070611726268189} a^{27} - \frac{8809185300087726050}{891070611726268189} a^{26} + \frac{15898363123085325575}{891070611726268189} a^{25} - \frac{29017700027360344989}{891070611726268189} a^{24} + \frac{52720882541878669475}{891070611726268189} a^{23} - \frac{99448272604604810950}{891070611726268189} a^{22} + \frac{120562847150952571050}{891070611726268189} a^{21} - \frac{195478381529692945775}{891070611726268189} a^{20} + \frac{234045881310906218400}{891070611726268189} a^{19} - \frac{335317174544586589950}{891070611726268189} a^{18} + \frac{351260496660577901910}{891070611726268189} a^{17} - \frac{379990544788936500475}{891070611726268189} a^{16} + \frac{98052912636802113025}{891070611726268189} a^{15} - \frac{532674765695347151275}{891070611726268189} a^{14} + \frac{216408781046733414650}{891070611726268189} a^{13} - \frac{806227751598000450075}{891070611726268189} a^{12} + \frac{396987714866533122850}{891070611726268189} a^{11} - \frac{1041664400724339229980}{891070611726268189} a^{10} + \frac{198244128579968539425}{891070611726268189} a^{9} - \frac{157788054345688975425}{891070611726268189} a^{8} + \frac{69184257195550769825}{891070611726268189} a^{7} - \frac{62853630764445912125}{891070611726268189} a^{6} - \frac{15111717235196556275}{891070611726268189} a^{5} - \frac{3671014143480923400}{891070611726268189} a^{4} - \frac{64432239260792924670}{891070611726268189} a^{3} - \frac{221634357301994525}{891070611726268189} a^{2} - \frac{43702549327153850}{891070611726268189} a - \frac{15608053331126375}{891070611726268189} \) (order $14$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 4866030378143.887 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6^2$ (as 36T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 36
The 36 conjugacy class representatives for $C_6^2$
Character table for $C_6^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-91}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{13}) \), 3.3.169.1, \(\Q(\zeta_{7})^+\), 3.3.8281.2, 3.3.8281.1, \(\Q(\sqrt{-7}, \sqrt{13})\), 6.0.127353499.1, 6.0.36924979.1, 6.0.6240321451.2, 6.0.6240321451.1, 6.0.9796423.1, \(\Q(\zeta_{13})^+\), \(\Q(\zeta_{7})\), 6.6.5274997.1, 6.0.480024727.2, 6.6.891474493.2, 6.0.480024727.1, 6.6.891474493.1, 9.9.567869252041.1, 12.0.16218913707543001.1, 12.0.1363454074150441.1, 12.0.38941611811810745401.1, 12.0.38941611811810745401.2, 18.0.243008175525757569678159896851.1, 18.0.110609092182866440454328583.1, 18.18.708478645847689707516501157.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{6}$

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

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
7Data not computed
$13$13.12.10.1$x^{12} - 117 x^{6} + 10816$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
13.12.10.1$x^{12} - 117 x^{6} + 10816$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
13.12.10.1$x^{12} - 117 x^{6} + 10816$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$