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Properties

Label 12.4.10713818144...7009.1
Degree $12$
Signature $[4, 4]$
Discriminant $3^{12}\cdot 17^{10}$
Root discriminant $31.80$
Ramified primes $3, 17$
Class number $1$
Class group Trivial
Galois group $F_9$ (as 12T46)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, -180, 549, -573, 447, -513, 169, -171, 54, -19, 12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 12*x^10 - 19*x^9 + 54*x^8 - 171*x^7 + 169*x^6 - 513*x^5 + 447*x^4 - 573*x^3 + 549*x^2 - 180*x + 16)
 
gp: K = bnfinit(x^12 + 12*x^10 - 19*x^9 + 54*x^8 - 171*x^7 + 169*x^6 - 513*x^5 + 447*x^4 - 573*x^3 + 549*x^2 - 180*x + 16, 1)
 

Normalized defining polynomial

\( x^{12} + 12 x^{10} - 19 x^{9} + 54 x^{8} - 171 x^{7} + 169 x^{6} - 513 x^{5} + 447 x^{4} - 573 x^{3} + 549 x^{2} - 180 x + 16 \)

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:  \(1071381814448517009=3^{12}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.80$
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:  $3, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{26} a^{8} + \frac{5}{26} a^{7} + \frac{4}{13} a^{6} - \frac{9}{26} a^{5} + \frac{4}{13} a^{4} + \frac{3}{13} a^{3} + \frac{1}{13} a^{2} + \frac{11}{26} a - \frac{3}{13}$, $\frac{1}{52} a^{9} + \frac{9}{52} a^{7} - \frac{23}{52} a^{6} - \frac{25}{52} a^{5} + \frac{9}{26} a^{4} + \frac{6}{13} a^{3} + \frac{1}{52} a^{2} - \frac{9}{52} a + \frac{1}{13}$, $\frac{1}{52} a^{10} - \frac{1}{52} a^{8} - \frac{21}{52} a^{7} - \frac{1}{52} a^{6} + \frac{1}{13} a^{5} - \frac{1}{13} a^{4} - \frac{7}{52} a^{3} + \frac{23}{52} a^{2} - \frac{1}{26} a + \frac{2}{13}$, $\frac{1}{104} a^{11} - \frac{1}{104} a^{10} - \frac{1}{104} a^{9} + \frac{2}{13} a^{7} + \frac{9}{104} a^{6} - \frac{4}{13} a^{5} + \frac{1}{104} a^{4} - \frac{3}{52} a^{3} - \frac{37}{104} a^{2} - \frac{15}{52} a - \frac{3}{13}$

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:  \( 37838.1143405 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_9$ (as 12T46):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 9 conjugacy class representatives for $F_9$
Character table for $F_9$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1

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

Sibling fields

Degree 9 sibling: data not computed
Degree 18 sibling: data not computed
Degree 24 sibling: data not computed
Degree 36 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$3$3.12.12.4$x^{12} + 33 x^{11} - 24 x^{10} + 66 x^{9} - 45 x^{8} + 99 x^{7} - 99 x^{6} + 81 x^{5} + 27 x^{4} + 27 x^{3} + 81 x^{2} + 81$$3$$4$$12$12T46$[3/2, 3/2]_{2}^{4}$
$17$17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$