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Properties

Label 12.0.33270205387539712.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{8}\cdot 37^{9}$
Root discriminant $23.81$
Ramified primes $2, 37$
Class number $4$
Class group $[2, 2]$
Galois group $C_3 : C_4$ (as 12T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![343, 931, 1463, 1475, 1161, 486, 179, 54, 39, -13, 14, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 14*x^10 - 13*x^9 + 39*x^8 + 54*x^7 + 179*x^6 + 486*x^5 + 1161*x^4 + 1475*x^3 + 1463*x^2 + 931*x + 343)
 
gp: K = bnfinit(x^12 - 2*x^11 + 14*x^10 - 13*x^9 + 39*x^8 + 54*x^7 + 179*x^6 + 486*x^5 + 1161*x^4 + 1475*x^3 + 1463*x^2 + 931*x + 343, 1)
 

Normalized defining polynomial

\( x^{12} - 2 x^{11} + 14 x^{10} - 13 x^{9} + 39 x^{8} + 54 x^{7} + 179 x^{6} + 486 x^{5} + 1161 x^{4} + 1475 x^{3} + 1463 x^{2} + 931 x + 343 \)

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:  \(33270205387539712=2^{8}\cdot 37^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.81$
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:  $2, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2443} a^{10} - \frac{478}{2443} a^{9} + \frac{171}{349} a^{8} - \frac{797}{2443} a^{7} - \frac{633}{2443} a^{6} - \frac{611}{2443} a^{5} - \frac{381}{2443} a^{4} - \frac{375}{2443} a^{3} + \frac{867}{2443} a^{2} - \frac{1010}{2443} a + \frac{163}{349}$, $\frac{1}{3833890308101} a^{11} - \frac{91963083}{3833890308101} a^{10} - \frac{204304845553}{547698615443} a^{9} + \frac{895762009304}{3833890308101} a^{8} - \frac{1052997433204}{3833890308101} a^{7} - \frac{832527778679}{3833890308101} a^{6} - \frac{899833666581}{3833890308101} a^{5} + \frac{360946494492}{3833890308101} a^{4} + \frac{1512952268246}{3833890308101} a^{3} + \frac{669818547704}{3833890308101} a^{2} - \frac{5486974908}{78242659349} a - \frac{28577164013}{78242659349}$

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:  $5$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 220.342128705 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:C_4$ (as 12T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 12
The 6 conjugacy class representatives for $C_3 : C_4$
Character table for $C_3 : C_4$

Intermediate fields

\(\Q(\sqrt{37}) \), 3.3.148.1 x3, 4.0.50653.1, 6.6.810448.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}$

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$2.12.8.1$x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
$37$37.4.3.2$x^{4} - 148$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.3.2$x^{4} - 148$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.3.2$x^{4} - 148$$4$$1$$3$$C_4$$[\ ]_{4}$