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Properties

Label 12.0.9934520342020096.3
Degree $12$
Signature $[0, 6]$
Discriminant $2^{26}\cdot 23^{6}$
Root discriminant $21.53$
Ramified primes $2, 23$
Class number $1$
Class group Trivial
Galois group $C_2\times S_5$ (as 12T123)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![52, 24, 88, -280, 388, -300, 148, -32, 1, -4, 8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 8*x^10 - 4*x^9 + x^8 - 32*x^7 + 148*x^6 - 300*x^5 + 388*x^4 - 280*x^3 + 88*x^2 + 24*x + 52)
 
gp: K = bnfinit(x^12 - 4*x^11 + 8*x^10 - 4*x^9 + x^8 - 32*x^7 + 148*x^6 - 300*x^5 + 388*x^4 - 280*x^3 + 88*x^2 + 24*x + 52, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} + 8 x^{10} - 4 x^{9} + x^{8} - 32 x^{7} + 148 x^{6} - 300 x^{5} + 388 x^{4} - 280 x^{3} + 88 x^{2} + 24 x + 52 \)

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:  \(9934520342020096=2^{26}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.53$
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:  $2, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{20} a^{10} - \frac{1}{20} a^{8} - \frac{2}{5} a^{7} - \frac{7}{20} a^{6} - \frac{2}{5} a^{5} + \frac{9}{20} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{1}{5} a - \frac{1}{10}$, $\frac{1}{51373800} a^{11} + \frac{137903}{8562300} a^{10} + \frac{1877077}{25686900} a^{9} + \frac{520973}{6421725} a^{8} + \frac{11317499}{51373800} a^{7} - \frac{2872109}{8562300} a^{6} + \frac{556111}{5137380} a^{5} - \frac{309662}{1284345} a^{4} + \frac{4353589}{25686900} a^{3} + \frac{761059}{12843450} a^{2} + \frac{260642}{1284345} a + \frac{671248}{6421725}$

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:  \( \frac{97663}{25686900} a^{11} - \frac{24548}{2140575} a^{10} - \frac{6262}{6421725} a^{9} + \frac{551351}{12843450} a^{8} - \frac{1264783}{25686900} a^{7} - \frac{415036}{2140575} a^{6} + \frac{442658}{1284345} a^{5} - \frac{449633}{2568690} a^{4} - \frac{13261883}{12843450} a^{3} + \frac{11250382}{6421725} a^{2} - \frac{409423}{256869} a - \frac{1456822}{6421725} \) (order $4$)
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:  \( 25182.4122752 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_5$ (as 12T123):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 240
The 14 conjugacy class representatives for $C_2\times S_5$
Character table for $C_2\times S_5$

Intermediate fields

\(\Q(\sqrt{-1}) \), 6.2.49836032.1

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

Sibling fields

Degree 10 siblings: data not computed
Degree 12 sibling: data not computed
Degree 20 siblings: data not computed
Degree 24 siblings: data not computed
Degree 30 siblings: data not computed
Degree 40 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ R ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$2$2.4.9.3$x^{4} + 6 x^{2} + 10$$4$$1$$9$$D_{4}$$[2, 3, 7/2]$
2.4.9.3$x^{4} + 6 x^{2} + 10$$4$$1$$9$$D_{4}$$[2, 3, 7/2]$
2.4.8.1$x^{4} + 2 x^{2} + 4 x + 10$$4$$1$$8$$C_2^2$$[2, 3]$
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.8.6.2$x^{8} - 1633 x^{4} + 1270129$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$