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Properties

Label 12.12.2027009431...5881.2
Degree $12$
Signature $[12, 0]$
Discriminant $7^{8}\cdot 181^{6}$
Root discriminant $49.23$
Ramified primes $7, 181$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $A_4$ (as 12T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3375, -9900, 53940, -53239, -4597, 25793, -6342, -3153, 1142, 120, -61, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 61*x^10 + 120*x^9 + 1142*x^8 - 3153*x^7 - 6342*x^6 + 25793*x^5 - 4597*x^4 - 53239*x^3 + 53940*x^2 - 9900*x - 3375)
 
gp: K = bnfinit(x^12 - x^11 - 61*x^10 + 120*x^9 + 1142*x^8 - 3153*x^7 - 6342*x^6 + 25793*x^5 - 4597*x^4 - 53239*x^3 + 53940*x^2 - 9900*x - 3375, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} - 61 x^{10} + 120 x^{9} + 1142 x^{8} - 3153 x^{7} - 6342 x^{6} + 25793 x^{5} - 4597 x^{4} - 53239 x^{3} + 53940 x^{2} - 9900 x - 3375 \)

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:  \(202700943101784875881=7^{8}\cdot 181^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.23$
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:  $7, 181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6} - \frac{3}{7} a^{5} - \frac{3}{7} a^{4} - \frac{3}{7} a^{3} - \frac{1}{7} a^{2} + \frac{2}{7} a - \frac{1}{7}$, $\frac{1}{14} a^{7} + \frac{1}{7} a^{5} + \frac{1}{7} a^{4} + \frac{2}{7} a^{3} + \frac{3}{7} a^{2} - \frac{1}{7} a - \frac{3}{14}$, $\frac{1}{14} a^{8} - \frac{3}{7} a^{5} - \frac{2}{7} a^{4} - \frac{1}{7} a^{3} - \frac{1}{2} a + \frac{1}{7}$, $\frac{1}{98} a^{9} - \frac{1}{49} a^{8} - \frac{1}{98} a^{7} + \frac{3}{49} a^{6} - \frac{1}{49} a^{5} + \frac{19}{49} a^{4} - \frac{11}{49} a^{3} - \frac{11}{98} a^{2} - \frac{3}{7} a + \frac{43}{98}$, $\frac{1}{120540} a^{10} + \frac{599}{120540} a^{9} - \frac{983}{60270} a^{8} + \frac{29}{1148} a^{7} - \frac{982}{30135} a^{6} + \frac{3501}{10045} a^{5} - \frac{297}{20090} a^{4} + \frac{42893}{120540} a^{3} - \frac{37747}{120540} a^{2} + \frac{6869}{30135} a + \frac{3597}{8036}$, $\frac{1}{48247340400} a^{11} - \frac{1453}{588382200} a^{10} - \frac{201702091}{48247340400} a^{9} - \frac{89050379}{3216489360} a^{8} + \frac{1072386467}{48247340400} a^{7} + \frac{1821041}{191457700} a^{6} + \frac{80916197}{164106600} a^{5} - \frac{21058831117}{48247340400} a^{4} - \frac{488852827}{3015458775} a^{3} + \frac{20070178001}{48247340400} a^{2} - \frac{905631227}{3216489360} a - \frac{27446639}{214432624}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 923852.467934 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_4$ (as 12T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 12
The 4 conjugacy class representatives for $A_4$
Character table for $A_4$

Intermediate fields

\(\Q(\zeta_{7})^+\), 4.4.1605289.1 x4, 6.6.78659161.1 x3

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

Sibling fields

Degree 4 sibling: data not computed
Degree 6 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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
$181$181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$