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Properties

Label 12.4.4882812500000000.1
Degree $12$
Signature $[4, 4]$
Discriminant $2^{8}\cdot 5^{19}$
Root discriminant $20.30$
Ramified primes $2, 5$
Class number $1$
Class group Trivial
Galois group $A_5:C_4$ (as 12T124)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-35, 68, 24, -65, 15, 30, -47, 14, 5, -5, 5, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 + 5*x^10 - 5*x^9 + 5*x^8 + 14*x^7 - 47*x^6 + 30*x^5 + 15*x^4 - 65*x^3 + 24*x^2 + 68*x - 35)
 
gp: K = bnfinit(x^12 - 3*x^11 + 5*x^10 - 5*x^9 + 5*x^8 + 14*x^7 - 47*x^6 + 30*x^5 + 15*x^4 - 65*x^3 + 24*x^2 + 68*x - 35, 1)
 

Normalized defining polynomial

\( x^{12} - 3 x^{11} + 5 x^{10} - 5 x^{9} + 5 x^{8} + 14 x^{7} - 47 x^{6} + 30 x^{5} + 15 x^{4} - 65 x^{3} + 24 x^{2} + 68 x - 35 \)

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:  \(4882812500000000=2^{8}\cdot 5^{19}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.30$
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:  $2, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{5} a^{6} + \frac{2}{5} a$, $\frac{1}{5} a^{7} + \frac{2}{5} a^{2}$, $\frac{1}{5} a^{8} + \frac{2}{5} a^{3}$, $\frac{1}{25} a^{9} + \frac{1}{25} a^{8} + \frac{2}{25} a^{7} - \frac{2}{25} a^{6} + \frac{2}{5} a^{5} + \frac{7}{25} a^{4} + \frac{2}{25} a^{3} + \frac{9}{25} a^{2} + \frac{1}{25} a + \frac{1}{5}$, $\frac{1}{25} a^{10} + \frac{1}{25} a^{8} + \frac{1}{25} a^{7} + \frac{2}{25} a^{6} - \frac{3}{25} a^{5} - \frac{1}{5} a^{4} + \frac{7}{25} a^{3} + \frac{2}{25} a^{2} + \frac{9}{25} a - \frac{1}{5}$, $\frac{1}{67225} a^{11} + \frac{39}{13445} a^{10} + \frac{969}{67225} a^{9} - \frac{1751}{67225} a^{8} + \frac{188}{67225} a^{7} + \frac{2281}{67225} a^{6} + \frac{4808}{13445} a^{5} - \frac{21092}{67225} a^{4} - \frac{32452}{67225} a^{3} - \frac{4229}{67225} a^{2} + \frac{7028}{67225} a - \frac{2947}{13445}$

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:  \( 2621.63953476 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_5:C_4$ (as 12T124):

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 $A_5:C_4$
Character table for $A_5:C_4$

Intermediate fields

6.2.31250000.1

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

Sibling fields

Degree 20 sibling: data not computed
Degree 24 siblings: data not computed
Degree 40 siblings: data not computed
Arithmetically equvalently 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 ${\href{/LocalNumberField/3.12.0.1}{12} }$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/13.12.0.1}{12} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.12.8.1$x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.10.19.2$x^{10} + 80$$10$$1$$19$$F_5$$[9/4]_{4}$