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Properties

Label 12.4.3749169810374656.1
Degree $12$
Signature $[4, 4]$
Discriminant $2^{22}\cdot 19^{7}$
Root discriminant $19.85$
Ramified primes $2, 19$
Class number $1$
Class group Trivial
Galois group 12T221

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, -16, -36, 36, 8, 32, -4, -26, 1, -8, 7, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 7*x^10 - 8*x^9 + x^8 - 26*x^7 - 4*x^6 + 32*x^5 + 8*x^4 + 36*x^3 - 36*x^2 - 16*x + 4)
 
gp: K = bnfinit(x^12 + 7*x^10 - 8*x^9 + x^8 - 26*x^7 - 4*x^6 + 32*x^5 + 8*x^4 + 36*x^3 - 36*x^2 - 16*x + 4, 1)
 

Normalized defining polynomial

\( x^{12} + 7 x^{10} - 8 x^{9} + x^{8} - 26 x^{7} - 4 x^{6} + 32 x^{5} + 8 x^{4} + 36 x^{3} - 36 x^{2} - 16 x + 4 \)

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:  \(3749169810374656=2^{22}\cdot 19^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.85$
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:  $2, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5}$, $\frac{1}{12} a^{9} - \frac{1}{6} a^{8} - \frac{1}{12} a^{7} + \frac{1}{6} a^{6} - \frac{1}{12} a^{5} + \frac{1}{3} a^{4} - \frac{1}{6} a^{3} + \frac{1}{3} a^{2} + \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{444} a^{10} - \frac{5}{148} a^{9} + \frac{31}{444} a^{8} + \frac{21}{148} a^{7} + \frac{1}{148} a^{6} - \frac{43}{444} a^{5} - \frac{3}{37} a^{4} + \frac{25}{74} a^{3} - \frac{19}{222} a^{2} - \frac{83}{222} a + \frac{44}{111}$, $\frac{1}{444} a^{11} - \frac{3}{148} a^{9} - \frac{16}{111} a^{8} + \frac{97}{444} a^{7} - \frac{6}{37} a^{6} - \frac{50}{111} a^{5} + \frac{32}{111} a^{4} + \frac{11}{74} a^{3} + \frac{1}{111} a^{2} - \frac{14}{37} a - \frac{43}{111}$

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:  \( 4908.97221252 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

12T221:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1536
The 40 conjugacy class representatives for [1/2.cD(4)^3]S(3)
Character table for [1/2.cD(4)^3]S(3) is not computed

Intermediate fields

3.1.152.1, 6.2.1755904.1

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

Sibling fields

Degree 12 siblings: data not computed
Degree 24 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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.12.0.1}{12} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/13.12.0.1}{12} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.12.0.1}{12} }$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$2.4.10.4$x^{4} + 6 x^{2} - 1$$4$$1$$10$$D_{4}$$[2, 3, 7/2]$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.9$x^{4} + 2 x^{3} + 6$$4$$1$$6$$D_{4}$$[2, 2]^{2}$
$19$19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.8.7.2$x^{8} - 19$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$