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Properties

Label 12.0.61274422532...0881.1
Degree $12$
Signature $[0, 6]$
Discriminant $7^{10}\cdot 167^{6}$
Root discriminant $65.40$
Ramified primes $7, 167$
Class number $64$ (GRH)
Class group $[2, 2, 16]$ (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![6023704, -1140204, 3313884, -592913, 723945, -110775, 77050, -8778, 4119, -307, 105, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 105*x^10 - 307*x^9 + 4119*x^8 - 8778*x^7 + 77050*x^6 - 110775*x^5 + 723945*x^4 - 592913*x^3 + 3313884*x^2 - 1140204*x + 6023704)
 
gp: K = bnfinit(x^12 - 4*x^11 + 105*x^10 - 307*x^9 + 4119*x^8 - 8778*x^7 + 77050*x^6 - 110775*x^5 + 723945*x^4 - 592913*x^3 + 3313884*x^2 - 1140204*x + 6023704, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} + 105 x^{10} - 307 x^{9} + 4119 x^{8} - 8778 x^{7} + 77050 x^{6} - 110775 x^{5} + 723945 x^{4} - 592913 x^{3} + 3313884 x^{2} - 1140204 x + 6023704 \)

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:  \(6127442253232770770881=7^{10}\cdot 167^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $65.40$
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:  $7, 167$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{52} a^{8} + \frac{5}{26} a^{7} + \frac{3}{26} a^{6} - \frac{3}{13} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{9}{52} a^{2} - \frac{6}{13} a - \frac{4}{13}$, $\frac{1}{52} a^{9} + \frac{5}{26} a^{7} + \frac{3}{26} a^{6} + \frac{3}{52} a^{5} - \frac{17}{52} a^{3} + \frac{4}{13} a^{2} - \frac{5}{26} a + \frac{1}{13}$, $\frac{1}{52} a^{10} + \frac{5}{26} a^{7} - \frac{5}{52} a^{6} - \frac{5}{26} a^{5} + \frac{9}{52} a^{4} - \frac{5}{26} a^{3} - \frac{11}{26} a^{2} - \frac{4}{13} a + \frac{1}{13}$, $\frac{1}{67441636950724935834988} a^{11} + \frac{6766758968145663458}{1296954556744710304519} a^{10} + \frac{77136268647996259296}{16860409237681233958747} a^{9} + \frac{29396502993660201424}{16860409237681233958747} a^{8} + \frac{884455960581667094499}{67441636950724935834988} a^{7} + \frac{95231013218500773483}{406274921389909252018} a^{6} - \frac{9559506988367992969237}{67441636950724935834988} a^{5} - \frac{3094618163537347549566}{16860409237681233958747} a^{4} - \frac{535926164584263127182}{1296954556744710304519} a^{3} - \frac{35249827705944618201}{411229493601981316067} a^{2} - \frac{3420062581770407062895}{16860409237681233958747} a - \frac{6348048305577789232066}{16860409237681233958747}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{16}$, which has order $64$ (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:  \( 31826.2424652 \) (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

\(\Q(\sqrt{1169}) \), \(\Q(\zeta_{7})^+\), 6.0.400967.1, 6.6.78277980641.1, 6.0.468730423.3

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

Sibling fields

Degree 12 siblings: 12.0.6127442253232770770881.2, 12.0.219708209445758929.1, 12.0.7877952219361.1, 12.0.219708209445758929.2
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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$7$7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
167Data not computed