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Properties

Label 12.0.84917815205...0000.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{8}\cdot 5^{9}\cdot 19^{8}$
Root discriminant $37.79$
Ramified primes $2, 5, 19$
Class number $48$
Class group $[4, 12]$
Galois group $C_3 : C_4$ (as 12T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![136161, -230256, 202860, -106605, 45121, -13190, 3839, -780, 211, -15, 15, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 15*x^10 - 15*x^9 + 211*x^8 - 780*x^7 + 3839*x^6 - 13190*x^5 + 45121*x^4 - 106605*x^3 + 202860*x^2 - 230256*x + 136161)
 
gp: K = bnfinit(x^12 - x^11 + 15*x^10 - 15*x^9 + 211*x^8 - 780*x^7 + 3839*x^6 - 13190*x^5 + 45121*x^4 - 106605*x^3 + 202860*x^2 - 230256*x + 136161, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 15 x^{10} - 15 x^{9} + 211 x^{8} - 780 x^{7} + 3839 x^{6} - 13190 x^{5} + 45121 x^{4} - 106605 x^{3} + 202860 x^{2} - 230256 x + 136161 \)

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

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(8491781520500000000=2^{8}\cdot 5^{9}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.79$
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:  $2, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{369} a^{10} + \frac{17}{369} a^{9} - \frac{16}{123} a^{8} - \frac{47}{123} a^{7} - \frac{113}{369} a^{6} + \frac{46}{123} a^{5} + \frac{50}{369} a^{4} - \frac{113}{369} a^{3} - \frac{86}{369} a^{2} - \frac{4}{41} a$, $\frac{1}{309895687956794974039557} a^{11} - \frac{267746513838348932536}{309895687956794974039557} a^{10} - \frac{1239658922905206287302}{34432854217421663782173} a^{9} - \frac{48407450769233620351358}{103298562652264991346519} a^{8} - \frac{35773621829249685503234}{309895687956794974039557} a^{7} - \frac{2803283745134023102661}{11477618072473887927391} a^{6} - \frac{63017524658500616344054}{309895687956794974039557} a^{5} - \frac{104313357037638313385996}{309895687956794974039557} a^{4} - \frac{10415179159839952075295}{309895687956794974039557} a^{3} - \frac{19834795281145810511152}{103298562652264991346519} a^{2} - \frac{6989741927055922410628}{34432854217421663782173} a - \frac{26275094041882275012}{279941904206680193351}$

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

Class group and class number

$C_{4}\times C_{12}$, which has order $48$

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $5$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{717347590328656391}{34432854217421663782173} a^{11} + \frac{2598119664056560157}{103298562652264991346519} a^{10} - \frac{25520972637141392753}{103298562652264991346519} a^{9} + \frac{13172311310179129430}{34432854217421663782173} a^{8} - \frac{43294501387936117262}{11477618072473887927391} a^{7} + \frac{1719635678407357396079}{103298562652264991346519} a^{6} - \frac{845024267669564280046}{11477618072473887927391} a^{5} + \frac{24892672800359094032698}{103298562652264991346519} a^{4} - \frac{86585255018123708537971}{103298562652264991346519} a^{3} + \frac{193650205446856979756102}{103298562652264991346519} a^{2} - \frac{100432957344368510026240}{34432854217421663782173} a + \frac{758097512112100041514}{279941904206680193351} \) (order $10$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3242.18396587 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:C_4$ (as 12T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 12
The 6 conjugacy class representatives for $C_3 : C_4$
Character table for $C_3 : C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 3.3.7220.1 x3, \(\Q(\zeta_{5})\), 6.6.260642000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.12.8.1$x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$19$19.6.4.1$x^{6} + 57 x^{3} + 1444$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
19.6.4.1$x^{6} + 57 x^{3} + 1444$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$