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Properties

Label 12.0.14860063685...3321.1
Degree $12$
Signature $[0, 6]$
Discriminant $13^{10}\cdot 47^{6}$
Root discriminant $58.12$
Ramified primes $13, 47$
Class number $40$ (GRH)
Class group $[2, 20]$ (GRH)
Galois group $C_2^2 \times A_4$ (as 12T25)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![17861, 56386, 75456, 59399, 30732, 9677, 1655, 402, 247, 68, 4, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 4*x^10 + 68*x^9 + 247*x^8 + 402*x^7 + 1655*x^6 + 9677*x^5 + 30732*x^4 + 59399*x^3 + 75456*x^2 + 56386*x + 17861)
 
gp: K = bnfinit(x^12 - x^11 + 4*x^10 + 68*x^9 + 247*x^8 + 402*x^7 + 1655*x^6 + 9677*x^5 + 30732*x^4 + 59399*x^3 + 75456*x^2 + 56386*x + 17861, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 4 x^{10} + 68 x^{9} + 247 x^{8} + 402 x^{7} + 1655 x^{6} + 9677 x^{5} + 30732 x^{4} + 59399 x^{3} + 75456 x^{2} + 56386 x + 17861 \)

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:  \(1486006368571562353321=13^{10}\cdot 47^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.12$
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:  $13, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{15} a^{9} - \frac{1}{5} a^{8} + \frac{2}{15} a^{7} - \frac{4}{15} a^{6} + \frac{1}{3} a^{5} + \frac{1}{5} a^{3} + \frac{1}{15} a^{2} + \frac{1}{15} a - \frac{2}{15}$, $\frac{1}{15} a^{10} - \frac{7}{15} a^{8} + \frac{2}{15} a^{7} - \frac{7}{15} a^{6} + \frac{1}{5} a^{4} - \frac{1}{3} a^{3} + \frac{4}{15} a^{2} + \frac{1}{15} a - \frac{2}{5}$, $\frac{1}{1164050692668827655} a^{11} - \frac{11336268238331746}{388016897556275885} a^{10} + \frac{4637243853984389}{1164050692668827655} a^{9} + \frac{633195090307264}{232810138533765531} a^{8} - \frac{251474972557649041}{1164050692668827655} a^{7} - \frac{168355547304192481}{388016897556275885} a^{6} + \frac{77730877299548386}{388016897556275885} a^{5} - \frac{552797506365290459}{1164050692668827655} a^{4} + \frac{534641242779841822}{1164050692668827655} a^{3} + \frac{90306021487771535}{232810138533765531} a^{2} + \frac{84048706422244324}{388016897556275885} a + \frac{174641649383993802}{388016897556275885}$

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

Class group and class number

$C_{2}\times C_{20}$, which has order $40$ (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:  \( 7222.39514479 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times A_4$ (as 12T25):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 48
The 16 conjugacy class representatives for $C_2^2 \times A_4$
Character table for $C_2^2 \times A_4$

Intermediate fields

\(\Q(\sqrt{-47}) \), 3.3.169.1, 6.0.17450771.1, 6.6.820186237.1, 6.0.2965288703.1

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

Sibling fields

Degree 12 siblings: 12.0.672705463364220169.1, 12.0.672705463364220169.2, 12.0.304529408494441.1, 12.0.1486006368571562353321.2
Degree 16 sibling: data not computed
Degree 24 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$13$13.12.10.1$x^{12} - 117 x^{6} + 10816$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$47$47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$