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Properties

Label 12.0.20648722209...0000.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{4}\cdot 3^{12}\cdot 5^{7}\cdot 11\cdot 41^{4}$
Root discriminant $40.70$
Ramified primes $2, 3, 5, 11, 41$
Class number $1$
Class group Trivial
Galois group 12T274

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11344, 2592, 2088, -648, -1311, -216, 124, 72, 54, -8, -12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 12*x^10 - 8*x^9 + 54*x^8 + 72*x^7 + 124*x^6 - 216*x^5 - 1311*x^4 - 648*x^3 + 2088*x^2 + 2592*x + 11344)
 
gp: K = bnfinit(x^12 - 12*x^10 - 8*x^9 + 54*x^8 + 72*x^7 + 124*x^6 - 216*x^5 - 1311*x^4 - 648*x^3 + 2088*x^2 + 2592*x + 11344, 1)
 

Normalized defining polynomial

\( x^{12} - 12 x^{10} - 8 x^{9} + 54 x^{8} + 72 x^{7} + 124 x^{6} - 216 x^{5} - 1311 x^{4} - 648 x^{3} + 2088 x^{2} + 2592 x + 11344 \)

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:  \(20648722209513750000=2^{4}\cdot 3^{12}\cdot 5^{7}\cdot 11\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.70$
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:  $2, 3, 5, 11, 41$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{3} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{1}{8} a^{3} + \frac{1}{8} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{32} a^{6} + \frac{1}{16} a^{4} + \frac{1}{32} a^{2} + \frac{3}{8}$, $\frac{1}{32} a^{7} - \frac{1}{16} a^{5} - \frac{1}{8} a^{4} - \frac{3}{32} a^{3} + \frac{1}{8} a^{2} - \frac{3}{8} a$, $\frac{1}{64} a^{8} - \frac{1}{64} a^{7} - \frac{1}{32} a^{5} - \frac{3}{64} a^{4} - \frac{1}{64} a^{3} + \frac{5}{32} a^{2} - \frac{3}{16} a - \frac{3}{8}$, $\frac{1}{128} a^{9} - \frac{1}{128} a^{7} - \frac{1}{64} a^{6} - \frac{5}{128} a^{5} - \frac{1}{32} a^{4} + \frac{9}{128} a^{3} - \frac{1}{64} a^{2} - \frac{9}{32} a - \frac{3}{16}$, $\frac{1}{256} a^{10} + \frac{1}{256} a^{8} + \frac{3}{256} a^{6} - \frac{1}{16} a^{5} - \frac{29}{256} a^{4} + \frac{1}{16} a^{3} + \frac{5}{32} a^{2} - \frac{1}{4} a + \frac{7}{16}$, $\frac{1}{512} a^{11} - \frac{1}{512} a^{10} + \frac{1}{512} a^{9} - \frac{1}{512} a^{8} + \frac{3}{512} a^{7} - \frac{3}{512} a^{6} - \frac{13}{512} a^{5} - \frac{51}{512} a^{4} + \frac{3}{64} a^{3} + \frac{5}{64} a^{2} - \frac{5}{32} a - \frac{11}{32}$

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:  \( 103021.667339 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

12T274:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10368
The 54 conjugacy class representatives for [S(3)^4]D(4)=S(3)wrD(4) are not computed
Character table for [S(3)^4]D(4)=S(3)wrD(4) is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.1025.1

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

Sibling fields

Degree 12 sibling: data not computed
Degree 18 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 R R ${\href{/LocalNumberField/7.12.0.1}{12} }$ R ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.4.4$x^{4} - 5$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
$3$3.12.12.2$x^{12} + 78 x^{10} - 75 x^{9} - 36 x^{8} + 36 x^{7} + 81 x^{6} - 81 x^{5} - 81 x^{4} + 81 x^{3} + 81 x^{2} + 81 x - 81$$3$$4$$12$12T173$[3/2, 3/2, 3/2, 3/2]_{2}^{4}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
11.6.0.1$x^{6} + x^{2} - 2 x + 8$$1$$6$$0$$C_6$$[\ ]^{6}$
$41$41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.4.3.4$x^{4} + 8856$$4$$1$$3$$C_4$$[\ ]_{4}$
41.6.0.1$x^{6} - x + 7$$1$$6$$0$$C_6$$[\ ]^{6}$