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Properties

Label 12.0.40000000000000000.3
Degree $12$
Signature $[0, 6]$
Discriminant $2^{18}\cdot 5^{16}$
Root discriminant $24.18$
Ramified primes $2, 5$
Class number $1$
Class group Trivial
Galois group $C_2\times A_5$ (as 12T75)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5, -10, 10, 0, -20, 36, -22, 2, 25, -30, 18, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 18*x^10 - 30*x^9 + 25*x^8 + 2*x^7 - 22*x^6 + 36*x^5 - 20*x^4 + 10*x^2 - 10*x + 5)
 
gp: K = bnfinit(x^12 - 6*x^11 + 18*x^10 - 30*x^9 + 25*x^8 + 2*x^7 - 22*x^6 + 36*x^5 - 20*x^4 + 10*x^2 - 10*x + 5, 1)
 

Normalized defining polynomial

\( x^{12} - 6 x^{11} + 18 x^{10} - 30 x^{9} + 25 x^{8} + 2 x^{7} - 22 x^{6} + 36 x^{5} - 20 x^{4} + 10 x^{2} - 10 x + 5 \)

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:  \(40000000000000000=2^{18}\cdot 5^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.18$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{591} a^{11} + \frac{22}{591} a^{10} + \frac{43}{591} a^{9} - \frac{8}{591} a^{8} - \frac{199}{591} a^{7} - \frac{251}{591} a^{6} + \frac{14}{197} a^{5} + \frac{10}{197} a^{4} + \frac{229}{591} a^{3} - \frac{89}{591} a^{2} - \frac{118}{591} a + \frac{232}{591}$

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:  \( \frac{175}{591} a^{11} - \frac{287}{591} a^{10} - \frac{1340}{591} a^{9} + \frac{7465}{591} a^{8} - \frac{14140}{591} a^{7} + \frac{9265}{591} a^{6} + \frac{3238}{197} a^{5} - \frac{4160}{197} a^{4} + \frac{10525}{591} a^{3} - \frac{800}{591} a^{2} - \frac{5875}{591} a + \frac{3958}{591} \) (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:  \( 7586.51776622 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times A_5$ (as 12T75):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 120
The 10 conjugacy class representatives for $C_2\times A_5$
Character table for $C_2\times A_5$

Intermediate fields

\(\Q(\sqrt{-1}) \), 6.2.25000000.1

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

Sibling fields

Degree 10 sibling: data not computed
Degree 12 sibling: data not computed
Degree 20 siblings: data not computed
Degree 24 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 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 R ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ R ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.12.18.51$x^{12} + 10 x^{11} + 16 x^{10} + 16 x^{9} - 6 x^{8} + 16 x^{7} - 8 x^{6} - 8 x^{5} + 4 x^{4} - 8 x^{3} + 16 x + 8$$4$$3$$18$$A_4 \times C_2$$[2, 2, 2]^{3}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.5.8.6$x^{5} + 5 x^{4} + 5$$5$$1$$8$$D_{5}$$[2]^{2}$
5.5.8.6$x^{5} + 5 x^{4} + 5$$5$$1$$8$$D_{5}$$[2]^{2}$