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Properties

Label 12.2.47216136328...672.16
Degree $12$
Signature $[2, 5]$
Discriminant $-\,2^{18}\cdot 23^{9}$
Root discriminant $29.71$
Ramified primes $2, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_6\times C_2):C_2$ (as 12T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![211, -26, 581, 1796, 561, -306, 311, 302, 9, 28, 15, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 15*x^10 + 28*x^9 + 9*x^8 + 302*x^7 + 311*x^6 - 306*x^5 + 561*x^4 + 1796*x^3 + 581*x^2 - 26*x + 211)
 
gp: K = bnfinit(x^12 - 2*x^11 + 15*x^10 + 28*x^9 + 9*x^8 + 302*x^7 + 311*x^6 - 306*x^5 + 561*x^4 + 1796*x^3 + 581*x^2 - 26*x + 211, 1)
 

Normalized defining polynomial

\( x^{12} - 2 x^{11} + 15 x^{10} + 28 x^{9} + 9 x^{8} + 302 x^{7} + 311 x^{6} - 306 x^{5} + 561 x^{4} + 1796 x^{3} + 581 x^{2} - 26 x + 211 \)

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:  \(-472161363286556672=-\,2^{18}\cdot 23^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.71$
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:  $2, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{7} a^{9} + \frac{1}{7} a^{8} + \frac{2}{7} a^{7} + \frac{1}{7} a^{6} - \frac{2}{7} a^{5} + \frac{1}{7} a^{4} - \frac{1}{7} a^{2} + \frac{2}{7} a - \frac{2}{7}$, $\frac{1}{2065} a^{10} - \frac{79}{2065} a^{9} - \frac{491}{2065} a^{8} + \frac{226}{2065} a^{7} - \frac{54}{2065} a^{6} + \frac{138}{295} a^{5} + \frac{61}{413} a^{4} - \frac{108}{413} a^{3} - \frac{149}{2065} a^{2} + \frac{16}{35} a - \frac{421}{2065}$, $\frac{1}{64433778941735} a^{11} - \frac{990395834}{12886755788347} a^{10} - \frac{939263147377}{64433778941735} a^{9} - \frac{19017516486523}{64433778941735} a^{8} - \frac{2657688054159}{12886755788347} a^{7} - \frac{5350944084714}{12886755788347} a^{6} - \frac{29340495440441}{64433778941735} a^{5} + \frac{2874818211492}{12886755788347} a^{4} + \frac{2571348557016}{64433778941735} a^{3} - \frac{393639956363}{1092097948165} a^{2} - \frac{134026310398}{12886755788347} a - \frac{95348187801}{1092097948165}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $6$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 21667.8214484 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:D_4$ (as 12T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 24
The 9 conjugacy class representatives for $(C_6\times C_2):C_2$
Character table for $(C_6\times C_2):C_2$

Intermediate fields

\(\Q(\sqrt{46}) \), 3.1.23.1, 4.2.778688.2, 6.2.6229504.1

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

Sibling fields

Galois closure: data not computed
Degree 12 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 R ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{12}$

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
$2$2.6.9.5$x^{6} - 4 x^{4} + 4 x^{2} + 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.5$x^{6} - 4 x^{4} + 4 x^{2} + 8$$2$$3$$9$$C_6$$[3]^{3}$
$23$23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.8.6.2$x^{8} - 1633 x^{4} + 1270129$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$