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Properties

Label 12.12.2425397972...0048.1
Degree $12$
Signature $[12, 0]$
Discriminant $2^{8}\cdot 3^{6}\cdot 37^{9}$
Root discriminant $41.25$
Ramified primes $2, 3, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
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, -2695, -609, 21899, 17219, -8838, -9515, 54, 1223, 61, -60, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 - 60*x^10 + 61*x^9 + 1223*x^8 + 54*x^7 - 9515*x^6 - 8838*x^5 + 17219*x^4 + 21899*x^3 - 609*x^2 - 2695*x + 343)
 
gp: K = bnfinit(x^12 - 2*x^11 - 60*x^10 + 61*x^9 + 1223*x^8 + 54*x^7 - 9515*x^6 - 8838*x^5 + 17219*x^4 + 21899*x^3 - 609*x^2 - 2695*x + 343, 1)
 

Normalized defining polynomial

\( x^{12} - 2 x^{11} - 60 x^{10} + 61 x^{9} + 1223 x^{8} + 54 x^{7} - 9515 x^{6} - 8838 x^{5} + 17219 x^{4} + 21899 x^{3} - 609 x^{2} - 2695 x + 343 \)

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

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(24253979727516450048=2^{8}\cdot 3^{6}\cdot 37^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.25$
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:  $2, 3, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{32921} a^{10} + \frac{14243}{32921} a^{9} - \frac{14011}{32921} a^{8} + \frac{481}{32921} a^{7} + \frac{12766}{32921} a^{6} - \frac{11888}{32921} a^{5} - \frac{3887}{32921} a^{4} + \frac{9488}{32921} a^{3} - \frac{7204}{32921} a^{2} + \frac{6303}{32921} a + \frac{2301}{4703}$, $\frac{1}{20435465416683457} a^{11} - \frac{227025028274}{20435465416683457} a^{10} + \frac{8910383401525119}{20435465416683457} a^{9} + \frac{7501610365935103}{20435465416683457} a^{8} + \frac{4635437281928934}{20435465416683457} a^{7} + \frac{9833759614750629}{20435465416683457} a^{6} + \frac{8863521473182808}{20435465416683457} a^{5} - \frac{2757412742335472}{20435465416683457} a^{4} - \frac{4167365229461414}{20435465416683457} a^{3} + \frac{9972473206498645}{20435465416683457} a^{2} + \frac{351042183063325}{2919352202383351} a - \frac{139654113368600}{417050314626193}$

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:  $11$
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:  \( 316765.282891 \) (assuming GRH)
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, 4.4.455877.1, 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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}$

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$$[\ ]_{3}^{4}$
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$37$37.4.3.2$x^{4} - 148$$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}$