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Properties

Label 12.4.16681088000000000.1
Degree $12$
Signature $[4, 4]$
Discriminant $2^{16}\cdot 5^{9}\cdot 19^{4}$
Root discriminant $22.48$
Ramified primes $2, 5, 19$
Class number $1$
Class group Trivial
Galois group 12T159

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![356, 248, -212, -52, 72, -24, -98, 4, 42, 6, -10, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 - 10*x^10 + 6*x^9 + 42*x^8 + 4*x^7 - 98*x^6 - 24*x^5 + 72*x^4 - 52*x^3 - 212*x^2 + 248*x + 356)
 
gp: K = bnfinit(x^12 - 2*x^11 - 10*x^10 + 6*x^9 + 42*x^8 + 4*x^7 - 98*x^6 - 24*x^5 + 72*x^4 - 52*x^3 - 212*x^2 + 248*x + 356, 1)
 

Normalized defining polynomial

\( x^{12} - 2 x^{11} - 10 x^{10} + 6 x^{9} + 42 x^{8} + 4 x^{7} - 98 x^{6} - 24 x^{5} + 72 x^{4} - 52 x^{3} - 212 x^{2} + 248 x + 356 \)

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

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(16681088000000000=2^{16}\cdot 5^{9}\cdot 19^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.48$
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:  $2, 5, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{1892876102278} a^{11} + \frac{76869637766}{946438051139} a^{10} + \frac{50939097390}{946438051139} a^{9} + \frac{43588643964}{946438051139} a^{8} + \frac{89667558327}{946438051139} a^{7} + \frac{123300661324}{946438051139} a^{6} - \frac{449217411155}{946438051139} a^{5} + \frac{412096357035}{946438051139} a^{4} + \frac{466707918022}{946438051139} a^{3} - \frac{183145096005}{946438051139} a^{2} + \frac{6095950177}{135205435877} a - \frac{417470842700}{946438051139}$

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:  $7$
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:  \( 2860.25496045 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

12T159:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 576
The 28 conjugacy class representatives for [2^5]F_18(6)_2
Character table for [2^5]F_18(6)_2 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 6.6.722000.1

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

Sibling fields

Degree 16 sibling: data not computed
Degree 24 siblings: data not computed
Degree 36 siblings: 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 ${\href{/LocalNumberField/3.12.0.1}{12} }$ R ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.12.0.1}{12} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }$ R ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.12.16.12$x^{12} + 18 x^{10} + 171 x^{8} - 404 x^{6} - 281 x^{4} - 286 x^{2} + 461$$6$$2$$16$12T159$[4/3, 4/3, 4/3, 4/3, 2]_{3}^{6}$
$5$5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$19$19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
19.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$