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Properties

Label 12.2.190459624995843007.1
Degree $12$
Signature $[2, 5]$
Discriminant $-\,7^{8}\cdot 127^{5}$
Root discriminant $27.54$
Ramified primes $7, 127$
Class number $4$
Class group $[2, 2]$
Galois group $(C_2^2.\SL(2,3)):C_2$ (as 12T104)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-811, -844, 51, 932, -112, -498, 563, -276, 91, -50, 22, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 22*x^10 - 50*x^9 + 91*x^8 - 276*x^7 + 563*x^6 - 498*x^5 - 112*x^4 + 932*x^3 + 51*x^2 - 844*x - 811)
 
gp: K = bnfinit(x^12 - 6*x^11 + 22*x^10 - 50*x^9 + 91*x^8 - 276*x^7 + 563*x^6 - 498*x^5 - 112*x^4 + 932*x^3 + 51*x^2 - 844*x - 811, 1)
 

Normalized defining polynomial

\( x^{12} - 6 x^{11} + 22 x^{10} - 50 x^{9} + 91 x^{8} - 276 x^{7} + 563 x^{6} - 498 x^{5} - 112 x^{4} + 932 x^{3} + 51 x^{2} - 844 x - 811 \)

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

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-190459624995843007=-\,7^{8}\cdot 127^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.54$
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:  $7, 127$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{58} a^{10} + \frac{6}{29} a^{9} + \frac{6}{29} a^{8} + \frac{3}{29} a^{7} + \frac{5}{29} a^{6} + \frac{27}{58} a^{5} + \frac{7}{58} a^{4} + \frac{11}{29} a^{3} - \frac{11}{29} a^{2} - \frac{14}{29} a - \frac{5}{58}$, $\frac{1}{29767689909998} a^{11} - \frac{138984069303}{29767689909998} a^{10} + \frac{4641230341347}{29767689909998} a^{9} + \frac{5961146255513}{29767689909998} a^{8} - \frac{2571190582395}{29767689909998} a^{7} + \frac{1954661683459}{29767689909998} a^{6} - \frac{5293943816131}{29767689909998} a^{5} - \frac{5860637405847}{14883844954999} a^{4} + \frac{2861221429599}{14883844954999} a^{3} + \frac{3036426910869}{29767689909998} a^{2} - \frac{1059115608447}{29767689909998} a - \frac{1220096078253}{29767689909998}$

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:  $6$
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:  \( 1250.41147298 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^2.\SL(2,3)):C_2$ (as 12T104):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 192
The 18 conjugacy class representatives for $(C_2^2.\SL(2,3)):C_2$
Character table for $(C_2^2.\SL(2,3)):C_2$

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.4.304927.1

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

Sibling fields

Degree 12 siblings: 12.6.11808520366783.2, 12.6.11808520366783.1
Degree 24 siblings: data not computed
Arithmetically equvalently sibling: 12.2.190459624995843007.2

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/3.12.0.1}{12} }$ ${\href{/LocalNumberField/5.12.0.1}{12} }$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }$ ${\href{/LocalNumberField/59.12.0.1}{12} }$

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$7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
127Data not computed