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Properties

Label 12.0.57018572461...3696.2
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 7^{8}\cdot 701^{4}$
Root discriminant $65.01$
Ramified primes $2, 7, 701$
Class number $1344$ (GRH)
Class group $[2, 672]$ (GRH)
Galois group $C_2\times C_4:D_4:C_3$ (as 12T89)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![491401, 0, 844705, 0, 312364, 0, 43533, 0, 2819, 0, 86, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 86*x^10 + 2819*x^8 + 43533*x^6 + 312364*x^4 + 844705*x^2 + 491401)
 
gp: K = bnfinit(x^12 + 86*x^10 + 2819*x^8 + 43533*x^6 + 312364*x^4 + 844705*x^2 + 491401, 1)
 

Normalized defining polynomial

\( x^{12} + 86 x^{10} + 2819 x^{8} + 43533 x^{6} + 312364 x^{4} + 844705 x^{2} + 491401 \)

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:  \(5701857246143068573696=2^{12}\cdot 7^{8}\cdot 701^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $65.01$
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:  $2, 7, 701$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{5} a^{6} - \frac{2}{5} a^{4} - \frac{1}{5}$, $\frac{1}{5} a^{7} - \frac{2}{5} a^{5} - \frac{1}{5} a$, $\frac{1}{5} a^{8} + \frac{1}{5} a^{4} - \frac{1}{5} a^{2} - \frac{2}{5}$, $\frac{1}{5} a^{9} + \frac{1}{5} a^{5} - \frac{1}{5} a^{3} - \frac{2}{5} a$, $\frac{1}{49160530645} a^{10} - \frac{2751577031}{49160530645} a^{8} - \frac{216685406}{9832106129} a^{6} - \frac{3564932121}{9832106129} a^{4} - \frac{16037586236}{49160530645} a^{2} + \frac{24366878}{70129145}$, $\frac{1}{49160530645} a^{11} - \frac{2751577031}{49160530645} a^{9} - \frac{216685406}{9832106129} a^{7} - \frac{3564932121}{9832106129} a^{5} - \frac{16037586236}{49160530645} a^{3} + \frac{24366878}{70129145} a$

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

Class group and class number

$C_{2}\times C_{672}$, which has order $1344$ (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:  \( 332.002331818 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_4:D_4:C_3$ (as 12T89):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 192
The 20 conjugacy class representatives for $C_2\times C_4:D_4:C_3$
Character table for $C_2\times C_4:D_4:C_3$

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.6.1683101.1

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

Sibling fields

Degree 12 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.12.12.23$x^{12} - 2 x^{10} - 65 x^{8} + 100 x^{6} - 97 x^{4} - 98 x^{2} + 97$$2$$6$$12$$C_2^2 \times A_4$$[2, 2, 2]^{6}$
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
701Data not computed