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Properties

Label 12.0.24919973321128389.1
Degree $12$
Signature $[0, 6]$
Discriminant $3^{6}\cdot 7^{8}\cdot 181^{3}$
Root discriminant $23.25$
Ramified primes $3, 7, 181$
Class number $4$
Class group $[2, 2]$
Galois group $D_4\times A_4$ (as 12T51)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2149, -5243, 5096, -3158, 1551, -339, 19, -9, -15, -5, 2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 2*x^10 - 5*x^9 - 15*x^8 - 9*x^7 + 19*x^6 - 339*x^5 + 1551*x^4 - 3158*x^3 + 5096*x^2 - 5243*x + 2149)
 
gp: K = bnfinit(x^12 + 2*x^10 - 5*x^9 - 15*x^8 - 9*x^7 + 19*x^6 - 339*x^5 + 1551*x^4 - 3158*x^3 + 5096*x^2 - 5243*x + 2149, 1)
 

Normalized defining polynomial

\( x^{12} + 2 x^{10} - 5 x^{9} - 15 x^{8} - 9 x^{7} + 19 x^{6} - 339 x^{5} + 1551 x^{4} - 3158 x^{3} + 5096 x^{2} - 5243 x + 2149 \)

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:  \(24919973321128389=3^{6}\cdot 7^{8}\cdot 181^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.25$
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:  $3, 7, 181$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{13} a^{10} - \frac{6}{13} a^{9} + \frac{5}{13} a^{8} - \frac{6}{13} a^{7} - \frac{1}{13} a^{6} - \frac{1}{13} a^{3} - \frac{3}{13} a^{2} - \frac{5}{13}$, $\frac{1}{207303992063623} a^{11} - \frac{3002589233454}{207303992063623} a^{10} - \frac{51249587716471}{207303992063623} a^{9} + \frac{76974429072644}{207303992063623} a^{8} - \frac{33186082493987}{207303992063623} a^{7} - \frac{68809173963479}{207303992063623} a^{6} + \frac{503318560143}{2278065846853} a^{5} + \frac{71576088244595}{207303992063623} a^{4} + \frac{93236820101811}{207303992063623} a^{3} + \frac{14195645115269}{29614856009089} a^{2} - \frac{923734374}{213056518051} a + \frac{14529823931743}{29614856009089}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$

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:  \( \frac{3161823}{389500033} a^{11} + \frac{6431615}{389500033} a^{10} + \frac{18042369}{389500033} a^{9} + \frac{19454887}{389500033} a^{8} - \frac{11990477}{389500033} a^{7} - \frac{57477310}{389500033} a^{6} - \frac{4815297}{29961541} a^{5} - \frac{1221001745}{389500033} a^{4} + \frac{183324451}{29961541} a^{3} - \frac{4520842165}{389500033} a^{2} + \frac{5616870301}{389500033} a - \frac{2146917470}{389500033} \) (order $6$)
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:  \( 1105.23502312 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4\times A_4$ (as 12T51):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 96
The 20 conjugacy class representatives for $D_4\times A_4$
Character table for $D_4\times A_4$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\zeta_{7})^+\), 6.0.64827.1

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

Sibling fields

Degree 12 sibling: data not computed
Degree 16 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 sibling: 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 ${\href{/LocalNumberField/2.12.0.1}{12} }$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.12.0.1}{12} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$3$3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
181Data not computed