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Properties

Label 36.0.12354957928...2304.1
Degree $36$
Signature $[0, 18]$
Discriminant $2^{36}\cdot 3^{48}\cdot 7^{30}$
Root discriminant $43.80$
Ramified primes $2, 3, 7$
Class number $196$ (GRH)
Class group $[14, 14]$ (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, 0, -9, 0, 75, 0, -622, 0, 5157, 0, -42756, 0, 354484, 0, -246222, 0, 131850, 0, -63802, 0, 29340, 0, -13098, 0, 5644, 0, -1548, 0, 417, 0, -109, 0, 27, 0, -6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 6*x^34 + 27*x^32 - 109*x^30 + 417*x^28 - 1548*x^26 + 5644*x^24 - 13098*x^22 + 29340*x^20 - 63802*x^18 + 131850*x^16 - 246222*x^14 + 354484*x^12 - 42756*x^10 + 5157*x^8 - 622*x^6 + 75*x^4 - 9*x^2 + 1)
 
gp: K = bnfinit(x^36 - 6*x^34 + 27*x^32 - 109*x^30 + 417*x^28 - 1548*x^26 + 5644*x^24 - 13098*x^22 + 29340*x^20 - 63802*x^18 + 131850*x^16 - 246222*x^14 + 354484*x^12 - 42756*x^10 + 5157*x^8 - 622*x^6 + 75*x^4 - 9*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{36} - 6 x^{34} + 27 x^{32} - 109 x^{30} + 417 x^{28} - 1548 x^{26} + 5644 x^{24} - 13098 x^{22} + 29340 x^{20} - 63802 x^{18} + 131850 x^{16} - 246222 x^{14} + 354484 x^{12} - 42756 x^{10} + 5157 x^{8} - 622 x^{6} + 75 x^{4} - 9 x^{2} + 1 \)

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:  \(123549579287202724195633555037990063416945072951206088802304=2^{36}\cdot 3^{48}\cdot 7^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.80$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(252=2^{2}\cdot 3^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{252}(1,·)$, $\chi_{252}(139,·)$, $\chi_{252}(13,·)$, $\chi_{252}(145,·)$, $\chi_{252}(19,·)$, $\chi_{252}(151,·)$, $\chi_{252}(25,·)$, $\chi_{252}(157,·)$, $\chi_{252}(31,·)$, $\chi_{252}(163,·)$, $\chi_{252}(37,·)$, $\chi_{252}(169,·)$, $\chi_{252}(43,·)$, $\chi_{252}(181,·)$, $\chi_{252}(55,·)$, $\chi_{252}(187,·)$, $\chi_{252}(61,·)$, $\chi_{252}(193,·)$, $\chi_{252}(67,·)$, $\chi_{252}(199,·)$, $\chi_{252}(73,·)$, $\chi_{252}(205,·)$, $\chi_{252}(79,·)$, $\chi_{252}(211,·)$, $\chi_{252}(85,·)$, $\chi_{252}(223,·)$, $\chi_{252}(97,·)$, $\chi_{252}(229,·)$, $\chi_{252}(103,·)$, $\chi_{252}(235,·)$, $\chi_{252}(109,·)$, $\chi_{252}(241,·)$, $\chi_{252}(115,·)$, $\chi_{252}(247,·)$, $\chi_{252}(121,·)$, $\chi_{252}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{4} a^{14} + \frac{1}{4}$, $\frac{1}{4} a^{15} + \frac{1}{4} a$, $\frac{1}{4} a^{16} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{17} + \frac{1}{4} a^{3}$, $\frac{1}{4} a^{18} + \frac{1}{4} a^{4}$, $\frac{1}{4} a^{19} + \frac{1}{4} a^{5}$, $\frac{1}{4} a^{20} + \frac{1}{4} a^{6}$, $\frac{1}{4} a^{21} + \frac{1}{4} a^{7}$, $\frac{1}{4} a^{22} + \frac{1}{4} a^{8}$, $\frac{1}{4} a^{23} + \frac{1}{4} a^{9}$, $\frac{1}{4} a^{24} + \frac{1}{4} a^{10}$, $\frac{1}{4} a^{25} + \frac{1}{4} a^{11}$, $\frac{1}{120907588} a^{26} - \frac{156215}{120907588} a^{24} + \frac{5010173}{60453794} a^{22} + \frac{434413}{30226897} a^{20} - \frac{2443030}{30226897} a^{18} + \frac{2746609}{120907588} a^{16} + \frac{2319495}{30226897} a^{14} - \frac{35303995}{120907588} a^{12} + \frac{55767481}{120907588} a^{10} - \frac{26643471}{60453794} a^{8} + \frac{8619925}{30226897} a^{6} - \frac{13776319}{30226897} a^{4} + \frac{43374401}{120907588} a^{2} + \frac{4964999}{30226897}$, $\frac{1}{120907588} a^{27} - \frac{156215}{120907588} a^{25} + \frac{5010173}{60453794} a^{23} + \frac{434413}{30226897} a^{21} - \frac{2443030}{30226897} a^{19} + \frac{2746609}{120907588} a^{17} + \frac{2319495}{30226897} a^{15} - \frac{35303995}{120907588} a^{13} + \frac{55767481}{120907588} a^{11} - \frac{26643471}{60453794} a^{9} + \frac{8619925}{30226897} a^{7} - \frac{13776319}{30226897} a^{5} + \frac{43374401}{120907588} a^{3} + \frac{4964999}{30226897} a$, $\frac{1}{483630352} a^{28} + \frac{17655623}{241815176} a^{14} + \frac{212609133}{483630352}$, $\frac{1}{483630352} a^{29} + \frac{17655623}{241815176} a^{15} + \frac{212609133}{483630352} a$, $\frac{1}{483630352} a^{30} + \frac{17655623}{241815176} a^{16} + \frac{212609133}{483630352} a^{2}$, $\frac{1}{483630352} a^{31} + \frac{17655623}{241815176} a^{17} + \frac{212609133}{483630352} a^{3}$, $\frac{1}{483630352} a^{32} + \frac{17655623}{241815176} a^{18} + \frac{212609133}{483630352} a^{4}$, $\frac{1}{483630352} a^{33} + \frac{17655623}{241815176} a^{19} + \frac{212609133}{483630352} a^{5}$, $\frac{1}{483630352} a^{34} + \frac{17655623}{241815176} a^{20} + \frac{212609133}{483630352} a^{6}$, $\frac{1}{483630352} a^{35} + \frac{17655623}{241815176} a^{21} + \frac{212609133}{483630352} a^{7}$

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

Class group and class number

$C_{14}\times C_{14}$, which has order $196$ (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:  \( -\frac{35525}{483630352} a^{33} - \frac{128801913}{241815176} a^{19} - \frac{95722144749}{483630352} a^{5} \) (order $28$)
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:  \( 57365817603378.39 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-7}) \), \(\Q(\zeta_{9})^+\), 3.3.3969.2, \(\Q(\zeta_{7})^+\), 3.3.3969.1, \(\Q(i, \sqrt{7})\), 6.0.419904.1, 6.0.1008189504.2, 6.0.153664.1, 6.0.1008189504.1, 6.6.144027072.1, 6.0.2250423.1, 6.6.7057326528.1, 6.0.110270727.1, \(\Q(\zeta_{28})^+\), \(\Q(\zeta_{7})\), 6.6.7057326528.2, 6.0.110270727.2, 9.9.62523502209.1, 12.0.20743797468893184.1, 12.0.49805857722812534784.1, \(\Q(\zeta_{28})\), 12.0.49805857722812534784.2, 18.0.1024770265180753855691096064.1, 18.18.351496200956998572502045949952.1, 18.0.1340851596668237962730583.1

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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{6}$

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
$2$2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
3Data not computed
7Data not computed