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Properties

Label 12.0.54959590533...2864.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 7^{10}\cdot 41^{6}$
Root discriminant $64.81$
Ramified primes $2, 7, 41$
Class number $2688$ (GRH)
Class group $[4, 4, 168]$ (GRH)
Galois group $C_6\times C_2$ (as 12T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2725801, 0, 1128523, 0, 200769, 0, 20259, 0, 1285, 0, 51, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 51*x^10 + 1285*x^8 + 20259*x^6 + 200769*x^4 + 1128523*x^2 + 2725801)
 
gp: K = bnfinit(x^12 + 51*x^10 + 1285*x^8 + 20259*x^6 + 200769*x^4 + 1128523*x^2 + 2725801, 1)
 

Normalized defining polynomial

\( x^{12} + 51 x^{10} + 1285 x^{8} + 20259 x^{6} + 200769 x^{4} + 1128523 x^{2} + 2725801 \)

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:  \(5495959053321957412864=2^{12}\cdot 7^{10}\cdot 41^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.81$
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:  $2, 7, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1148=2^{2}\cdot 7\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{1148}(1,·)$, $\chi_{1148}(1067,·)$, $\chi_{1148}(901,·)$, $\chi_{1148}(491,·)$, $\chi_{1148}(655,·)$, $\chi_{1148}(83,·)$, $\chi_{1148}(821,·)$, $\chi_{1148}(409,·)$, $\chi_{1148}(163,·)$, $\chi_{1148}(411,·)$, $\chi_{1148}(573,·)$, $\chi_{1148}(165,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{13} a^{7} + \frac{2}{13} a^{5} + \frac{1}{13} a^{3} + \frac{2}{13} a$, $\frac{1}{13} a^{8} + \frac{2}{13} a^{6} + \frac{1}{13} a^{4} + \frac{2}{13} a^{2}$, $\frac{1}{13} a^{9} - \frac{3}{13} a^{5} - \frac{4}{13} a$, $\frac{1}{590581823} a^{10} + \frac{21454112}{590581823} a^{8} - \frac{66344858}{590581823} a^{6} - \frac{18270001}{590581823} a^{4} + \frac{214716467}{590581823} a^{2} + \frac{1397248}{3494567}$, $\frac{1}{75003891521} a^{11} + \frac{2156634549}{75003891521} a^{9} + \frac{524236965}{75003891521} a^{7} - \frac{14691956834}{75003891521} a^{5} + \frac{27972062148}{75003891521} a^{3} + \frac{2722959082}{5769530117} a$

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

Class group and class number

$C_{4}\times C_{4}\times C_{168}$, which has order $2688$ (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:  $5$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
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:  \( 246.50546308257188 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_6$ (as 12T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 12
The 12 conjugacy class representatives for $C_6\times C_2$
Character table for $C_6\times C_2$

Intermediate fields

\(\Q(\sqrt{-41}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-287}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{7}, \sqrt{-41})\), 6.0.10590676544.4, \(\Q(\zeta_{28})^+\), 6.0.1158355247.1

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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.6.6.3$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}$
$7$7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
$41$41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$