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Properties

Label 36.0.49997585682...0625.3
Degree $36$
Signature $[0, 18]$
Discriminant $3^{54}\cdot 5^{18}\cdot 7^{30}$
Root discriminant $58.81$
Ramified primes $3, 5, 7$
Class number Not computed
Class group Not computed
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![594823321, 0, 0, 275010364, 0, 0, 90247619, 0, 0, 50229628, 0, 0, 8225230, 0, 0, -2654421, 0, 0, 415031, 0, 0, -33978, 0, 0, 2848, 0, 0, 1289, 0, 0, -70, 0, 0, -1, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - x^33 - 70*x^30 + 1289*x^27 + 2848*x^24 - 33978*x^21 + 415031*x^18 - 2654421*x^15 + 8225230*x^12 + 50229628*x^9 + 90247619*x^6 + 275010364*x^3 + 594823321)
 
gp: K = bnfinit(x^36 - x^33 - 70*x^30 + 1289*x^27 + 2848*x^24 - 33978*x^21 + 415031*x^18 - 2654421*x^15 + 8225230*x^12 + 50229628*x^9 + 90247619*x^6 + 275010364*x^3 + 594823321, 1)
 

Normalized defining polynomial

\( x^{36} - x^{33} - 70 x^{30} + 1289 x^{27} + 2848 x^{24} - 33978 x^{21} + 415031 x^{18} - 2654421 x^{15} + 8225230 x^{12} + 50229628 x^{9} + 90247619 x^{6} + 275010364 x^{3} + 594823321 \)

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:  \(4999758568289868528789868885747458073284974388119052886962890625=3^{54}\cdot 5^{18}\cdot 7^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.81$
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:  $3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(315=3^{2}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{315}(256,·)$, $\chi_{315}(1,·)$, $\chi_{315}(139,·)$, $\chi_{315}(269,·)$, $\chi_{315}(16,·)$, $\chi_{315}(19,·)$, $\chi_{315}(151,·)$, $\chi_{315}(281,·)$, $\chi_{315}(304,·)$, $\chi_{315}(34,·)$, $\chi_{315}(164,·)$, $\chi_{315}(296,·)$, $\chi_{315}(299,·)$, $\chi_{315}(46,·)$, $\chi_{315}(176,·)$, $\chi_{315}(116,·)$, $\chi_{315}(314,·)$, $\chi_{315}(59,·)$, $\chi_{315}(191,·)$, $\chi_{315}(194,·)$, $\chi_{315}(11,·)$, $\chi_{315}(199,·)$, $\chi_{315}(209,·)$, $\chi_{315}(211,·)$, $\chi_{315}(86,·)$, $\chi_{315}(89,·)$, $\chi_{315}(71,·)$, $\chi_{315}(221,·)$, $\chi_{315}(94,·)$, $\chi_{315}(226,·)$, $\chi_{315}(229,·)$, $\chi_{315}(104,·)$, $\chi_{315}(106,·)$, $\chi_{315}(244,·)$, $\chi_{315}(121,·)$, $\chi_{315}(124,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{13} a^{18} + \frac{5}{13} a^{15} + \frac{5}{13} a^{9} + \frac{5}{13} a^{3} - \frac{1}{13}$, $\frac{1}{13} a^{19} + \frac{5}{13} a^{16} + \frac{5}{13} a^{10} + \frac{5}{13} a^{4} - \frac{1}{13} a$, $\frac{1}{13} a^{20} + \frac{5}{13} a^{17} + \frac{5}{13} a^{11} + \frac{5}{13} a^{5} - \frac{1}{13} a^{2}$, $\frac{1}{13} a^{21} + \frac{1}{13} a^{15} + \frac{5}{13} a^{12} + \frac{1}{13} a^{9} + \frac{5}{13} a^{6} + \frac{5}{13}$, $\frac{1}{13} a^{22} + \frac{1}{13} a^{16} + \frac{5}{13} a^{13} + \frac{1}{13} a^{10} + \frac{5}{13} a^{7} + \frac{5}{13} a$, $\frac{1}{13} a^{23} + \frac{1}{13} a^{17} + \frac{5}{13} a^{14} + \frac{1}{13} a^{11} + \frac{5}{13} a^{8} + \frac{5}{13} a^{2}$, $\frac{1}{13} a^{24} + \frac{1}{13} a^{12} + \frac{1}{13}$, $\frac{1}{13} a^{25} + \frac{1}{13} a^{13} + \frac{1}{13} a$, $\frac{1}{13} a^{26} + \frac{1}{13} a^{14} + \frac{1}{13} a^{2}$, $\frac{1}{26} a^{27} - \frac{6}{13} a^{15} - \frac{1}{2} a^{9} - \frac{6}{13} a^{3} - \frac{1}{2}$, $\frac{1}{26} a^{28} - \frac{6}{13} a^{16} - \frac{1}{2} a^{10} - \frac{6}{13} a^{4} - \frac{1}{2} a$, $\frac{1}{26} a^{29} - \frac{6}{13} a^{17} - \frac{1}{2} a^{11} - \frac{6}{13} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4112401046} a^{30} - \frac{25393591}{4112401046} a^{27} + \frac{12696760}{2056200523} a^{24} + \frac{14910783}{2056200523} a^{21} - \frac{31936432}{2056200523} a^{18} + \frac{454998186}{2056200523} a^{15} - \frac{1362198399}{4112401046} a^{12} + \frac{1864147897}{4112401046} a^{9} - \frac{722929869}{2056200523} a^{6} - \frac{889789065}{4112401046} a^{3} + \frac{484278341}{4112401046}$, $\frac{1}{119259630334} a^{31} - \frac{408119973}{59629815167} a^{28} - \frac{1885334492}{59629815167} a^{25} + \frac{173080054}{59629815167} a^{22} + \frac{442571381}{59629815167} a^{19} + \frac{1562183083}{59629815167} a^{16} + \frac{16985437037}{119259630334} a^{13} + \frac{22996687253}{59629815167} a^{10} - \frac{16381687698}{59629815167} a^{7} - \frac{44228169319}{119259630334} a^{4} - \frac{12648656416}{59629815167} a$, $\frac{1}{3458529279686} a^{32} + \frac{3770668913}{3458529279686} a^{29} - \frac{1885334492}{1729264639843} a^{26} + \frac{9346897772}{1729264639843} a^{23} - \frac{54600334927}{1729264639843} a^{20} - \frac{649778874895}{1729264639843} a^{17} + \frac{1062800656889}{3458529279686} a^{14} - \frac{426458237971}{3458529279686} a^{11} + \frac{89117216059}{1729264639843} a^{8} + \frac{1496973207305}{3458529279686} a^{5} + \frac{951714274135}{3458529279686} a^{2}$, $\frac{1}{6973645287981629495856543618245300236606} a^{33} + \frac{51036087181267384091481883552}{3486822643990814747928271809122650118303} a^{30} - \frac{86926238240929785050756889811163454825}{6973645287981629495856543618245300236606} a^{27} + \frac{6647511320500414305427084939751359722}{3486822643990814747928271809122650118303} a^{24} + \frac{74766120849022868006383727049870262044}{3486822643990814747928271809122650118303} a^{21} - \frac{41448500183541055810188561541387575705}{3486822643990814747928271809122650118303} a^{18} - \frac{1386106997806534965219284602476782902587}{6973645287981629495856543618245300236606} a^{15} - \frac{800493848768396325112497377374705298343}{3486822643990814747928271809122650118303} a^{12} - \frac{3464089432409989571837831256201566822311}{6973645287981629495856543618245300236606} a^{9} - \frac{843127833454345736774598628208875805397}{6973645287981629495856543618245300236606} a^{6} + \frac{1132507690275387098414210575131388904649}{3486822643990814747928271809122650118303} a^{3} - \frac{125456187757004744073520089343065253}{285934039443258415509309263120476454}$, $\frac{1}{202235713351467255379839764929113706861574} a^{34} + \frac{51036087181267384091481883552}{101117856675733627689919882464556853430787} a^{31} - \frac{982223061733376555429143931977064451571}{101117856675733627689919882464556853430787} a^{28} + \frac{811298890702996125365797502429593694715}{101117856675733627689919882464556853430787} a^{25} + \frac{2220503132535678097500704840356116488692}{101117856675733627689919882464556853430787} a^{22} + \frac{2372505637963946077370922690928139429274}{101117856675733627689919882464556853430787} a^{19} + \frac{28654211165806638247701210983810664270485}{202235713351467255379839764929113706861574} a^{16} - \frac{20648561206869957197934967675457482894837}{101117856675733627689919882464556853430787} a^{13} - \frac{48535933283620161979263794912093279229915}{101117856675733627689919882464556853430787} a^{10} + \frac{27587887571393836054025156123098886697689}{202235713351467255379839764929113706861574} a^{7} - \frac{13887651391531199508046037218012334681887}{101117856675733627689919882464556853430787} a^{4} + \frac{239702140148020952059624752859740546}{4146043571927247024884984315246908583} a$, $\frac{1}{5864835687192550406015353182944297498985646} a^{35} + \frac{51036087181267384091481883552}{2932417843596275203007676591472148749492823} a^{32} + \frac{83596817217538624165227766529132439384447}{5864835687192550406015353182944297498985646} a^{29} + \frac{24146188892795371746116539609635021409512}{2932417843596275203007676591472148749492823} a^{26} + \frac{56668579804084554545919103090502114489885}{2932417843596275203007676591472148749492823} a^{23} + \frac{111268658981061698974207719191220135431660}{2932417843596275203007676591472148749492823} a^{20} - \frac{1822580395666855160998524329521153267770077}{5864835687192550406015353182944297498985646} a^{17} - \frac{129544714549967710094771764175749478897223}{2932417843596275203007676591472148749492823} a^{14} - \frac{1427160596686505734341319889934895938203259}{5864835687192550406015353182944297498985646} a^{11} - \frac{1450288479227789953260188510666578201906121}{5864835687192550406015353182944297498985646} a^{8} + \frac{1331757672062462461288580090964167330204740}{2932417843596275203007676591472148749492823} a^{5} - \frac{54694867869197322350119387920258610051}{240470527171780327443329090284320697814} a^{2}$

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

Class group and class number

Not computed

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:  \( \frac{6921108753981151158125696013}{8292087143854494049769968630493817166} a^{34} - \frac{941567415194979960768417862}{142967019721629207754654631560238227} a^{31} - \frac{150652800368259247394568166143}{4146043571927247024884984315246908583} a^{28} + \frac{6095073959946102864921771660252}{4146043571927247024884984315246908583} a^{25} - \frac{26254952032201136127001100887891}{4146043571927247024884984315246908583} a^{22} - \frac{105340678329434777534244437457400}{4146043571927247024884984315246908583} a^{19} + \frac{5039082461939296081592750135788093}{8292087143854494049769968630493817166} a^{16} - \frac{21832931462959167005202935547890502}{4146043571927247024884984315246908583} a^{13} + \frac{111571895622143247251603156194403821}{4146043571927247024884984315246908583} a^{10} - \frac{374570198742050087237942648772652019}{8292087143854494049769968630493817166} a^{7} - \frac{18185607450678345871428102267198351}{142967019721629207754654631560238227} a^{4} + \frac{2347059412181283559934910054110578416}{4146043571927247024884984315246908583} a \) (order $18$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{105}) \), \(\Q(\zeta_{9})^+\), 3.3.3969.2, 3.3.3969.1, \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{-3}, \sqrt{-35})\), \(\Q(\zeta_{9})\), 6.0.47258883.1, 6.0.47258883.2, 6.0.64827.1, 6.0.281302875.3, 6.6.843908625.1, 6.0.13783840875.2, 6.6.41351522625.2, 6.0.13783840875.1, 6.6.41351522625.1, 6.0.2100875.1, 6.6.56723625.1, 9.9.62523502209.1, 12.0.712181767349390625.1, 12.0.1709948423405886890625.5, 12.0.1709948423405886890625.6, 12.0.3217569633140625.2, 18.0.105548084868928352751387.1, 18.0.2618850774742652270958169921875.4, 18.18.70708970918051611315870587890625.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{6}$ R R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{6}$

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
3Data not computed
$5$5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
7Data not computed