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Properties

Label 12.0.22358026811...504.14
Degree $12$
Signature $[0, 6]$
Discriminant $2^{18}\cdot 31^{8}$
Root discriminant $27.91$
Ramified primes $2, 31$
Class number $1$
Class group Trivial
Galois group $A_5$ (as 12T33)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![31, -8, 94, -86, 61, -70, 92, -80, 46, -22, 10, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 10*x^10 - 22*x^9 + 46*x^8 - 80*x^7 + 92*x^6 - 70*x^5 + 61*x^4 - 86*x^3 + 94*x^2 - 8*x + 31)
 
gp: K = bnfinit(x^12 - 4*x^11 + 10*x^10 - 22*x^9 + 46*x^8 - 80*x^7 + 92*x^6 - 70*x^5 + 61*x^4 - 86*x^3 + 94*x^2 - 8*x + 31, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} + 10 x^{10} - 22 x^{9} + 46 x^{8} - 80 x^{7} + 92 x^{6} - 70 x^{5} + 61 x^{4} - 86 x^{3} + 94 x^{2} - 8 x + 31 \)

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:  \(223580268118933504=2^{18}\cdot 31^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.91$
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:  $2, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{11} a^{10} + \frac{1}{11} a^{9} - \frac{1}{11} a^{8} + \frac{1}{11} a^{7} + \frac{1}{11} a^{6} - \frac{3}{11} a^{5} - \frac{5}{11} a^{4} - \frac{3}{11} a^{3} + \frac{5}{11} a^{2} - \frac{2}{11} a + \frac{4}{11}$, $\frac{1}{67528549} a^{11} + \frac{633123}{67528549} a^{10} + \frac{16215644}{67528549} a^{9} + \frac{12565526}{67528549} a^{8} + \frac{17005281}{67528549} a^{7} - \frac{28306429}{67528549} a^{6} - \frac{9380306}{67528549} a^{5} + \frac{1870899}{6138959} a^{4} + \frac{25297578}{67528549} a^{3} - \frac{31893229}{67528549} a^{2} + \frac{14409376}{67528549} a + \frac{23780737}{67528549}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( 7108.11455748 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_5$ (as 12T33):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 60
The 5 conjugacy class representatives for $A_5$
Character table for $A_5$

Intermediate fields

6.2.59105344.4

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

Sibling fields

Degree 5 sibling: data not computed
Degree 6 sibling: data not computed
Degree 10 sibling: data not computed
Degree 15 sibling: data not computed
Degree 20 sibling: data not computed
Degree 30 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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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
$2$2.12.18.59$x^{12} - 2 x^{11} + 6 x^{10} + 4 x^{9} + 6 x^{8} + 12 x^{7} - 4 x^{6} - 8 x^{3} + 16 x^{2} - 8$$4$$3$$18$$A_4$$[2, 2]^{3}$
$31$$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
31.5.4.3$x^{5} - 1519$$5$$1$$4$$C_5$$[\ ]_{5}$
31.5.4.3$x^{5} - 1519$$5$$1$$4$$C_5$$[\ ]_{5}$