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Properties

Label 12.0.259729655939201.1
Degree $12$
Signature $[0, 6]$
Discriminant $7^{9}\cdot 23^{5}$
Root discriminant $15.89$
Ramified primes $7, 23$
Class number $1$
Class group Trivial
Galois group $D_{12}$ (as 12T12)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, -35, 53, -18, -30, 60, -49, 8, 21, -20, 13, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 13*x^10 - 20*x^9 + 21*x^8 + 8*x^7 - 49*x^6 + 60*x^5 - 30*x^4 - 18*x^3 + 53*x^2 - 35*x + 25)
 
gp: K = bnfinit(x^12 - 4*x^11 + 13*x^10 - 20*x^9 + 21*x^8 + 8*x^7 - 49*x^6 + 60*x^5 - 30*x^4 - 18*x^3 + 53*x^2 - 35*x + 25, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} + 13 x^{10} - 20 x^{9} + 21 x^{8} + 8 x^{7} - 49 x^{6} + 60 x^{5} - 30 x^{4} - 18 x^{3} + 53 x^{2} - 35 x + 25 \)

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:  \(259729655939201=7^{9}\cdot 23^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.89$
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:  $7, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{7} a^{8} + \frac{1}{7} a^{7} - \frac{1}{7} a^{6} + \frac{1}{7} a^{5} + \frac{2}{7} a^{4} - \frac{1}{7} a^{3} - \frac{1}{7} a^{2} - \frac{1}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{9} - \frac{2}{7} a^{7} + \frac{2}{7} a^{6} + \frac{1}{7} a^{5} - \frac{3}{7} a^{4} + \frac{2}{7} a - \frac{1}{7}$, $\frac{1}{35} a^{10} + \frac{2}{35} a^{9} - \frac{4}{35} a^{6} - \frac{6}{35} a^{5} + \frac{1}{7} a^{4} + \frac{1}{7} a^{3} - \frac{13}{35} a$, $\frac{1}{150535} a^{11} - \frac{692}{150535} a^{10} - \frac{5603}{150535} a^{9} + \frac{1950}{30107} a^{8} - \frac{2437}{8855} a^{7} - \frac{2494}{30107} a^{6} + \frac{33224}{150535} a^{5} + \frac{14976}{30107} a^{4} - \frac{300}{1309} a^{3} + \frac{58977}{150535} a^{2} + \frac{25317}{150535} a + \frac{1378}{4301}$

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:  \( 185.722134476 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{12}$ (as 12T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 24
The 9 conjugacy class representatives for $D_{12}$
Character table for $D_{12}$

Intermediate fields

\(\Q(\sqrt{-7}) \), 3.1.23.1, 4.0.7889.1, 6.0.181447.1

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

Sibling fields

Galois closure: data not computed
Degree 12 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/3.12.0.1}{12} }$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.12.0.1}{12} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.12.0.1}{12} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.12.0.1}{12} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}$

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
$7$7.4.3.1$x^{4} + 14$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
7.8.6.2$x^{8} - 49 x^{4} + 3969$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$