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Properties

Label 12.4.2193401363688512.1
Degree $12$
Signature $[4, 4]$
Discriminant $2^{6}\cdot 17^{11}$
Root discriminant $18.99$
Ramified primes $2, 17$
Class number $1$
Class group Trivial
Galois group $S_3^2:C_4$ (as 12T79)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, 2, -4, -15, -23, 11, 23, -15, 4, 2, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 + 2*x^10 + 4*x^9 - 15*x^8 + 23*x^7 + 11*x^6 - 23*x^5 - 15*x^4 - 4*x^3 + 2*x^2 + 3*x + 1)
 
gp: K = bnfinit(x^12 - 3*x^11 + 2*x^10 + 4*x^9 - 15*x^8 + 23*x^7 + 11*x^6 - 23*x^5 - 15*x^4 - 4*x^3 + 2*x^2 + 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{12} - 3 x^{11} + 2 x^{10} + 4 x^{9} - 15 x^{8} + 23 x^{7} + 11 x^{6} - 23 x^{5} - 15 x^{4} - 4 x^{3} + 2 x^{2} + 3 x + 1 \)

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

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2193401363688512=2^{6}\cdot 17^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.99$
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:  $2, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{86} a^{10} + \frac{12}{43} a^{9} - \frac{37}{86} a^{8} - \frac{25}{86} a^{7} - \frac{39}{86} a^{6} - \frac{23}{86} a^{5} + \frac{39}{86} a^{4} - \frac{25}{86} a^{3} + \frac{37}{86} a^{2} + \frac{12}{43} a - \frac{1}{86}$, $\frac{1}{1118} a^{11} + \frac{5}{1118} a^{10} + \frac{367}{1118} a^{9} - \frac{220}{559} a^{8} + \frac{46}{559} a^{7} + \frac{230}{559} a^{6} - \frac{20}{559} a^{5} + \frac{4}{559} a^{4} + \frac{213}{559} a^{3} - \frac{249}{1118} a^{2} - \frac{27}{1118} a - \frac{239}{1118}$

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

Class group and class number

Trivial group, which has order $1$

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1748.16673693 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3^2:C_4$ (as 12T79):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 144
The 18 conjugacy class representatives for $S_3^2:C_4$
Character table for $S_3^2:C_4$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 6.2.11358856.1

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 36 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/7.12.0.1}{12} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/31.12.0.1}{12} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.12.0.1}{12} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
$17$17.12.11.1$x^{12} - 17$$12$$1$$11$$S_3 \times C_4$$[\ ]_{12}^{2}$