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Properties

Label 12.4.56466369006...3328.5
Degree $12$
Signature $[4, 4]$
Discriminant $2^{27}\cdot 29^{10}$
Root discriminant $78.70$
Ramified primes $2, 29$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $A_4:C_4$ (as 12T30)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-386, 92, -1518, -1640, 427, -820, 1014, 740, 222, -64, -12, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 - 12*x^10 - 64*x^9 + 222*x^8 + 740*x^7 + 1014*x^6 - 820*x^5 + 427*x^4 - 1640*x^3 - 1518*x^2 + 92*x - 386)
 
gp: K = bnfinit(x^12 - 4*x^11 - 12*x^10 - 64*x^9 + 222*x^8 + 740*x^7 + 1014*x^6 - 820*x^5 + 427*x^4 - 1640*x^3 - 1518*x^2 + 92*x - 386, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} - 12 x^{10} - 64 x^{9} + 222 x^{8} + 740 x^{7} + 1014 x^{6} - 820 x^{5} + 427 x^{4} - 1640 x^{3} - 1518 x^{2} + 92 x - 386 \)

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:  \(56466369006718920163328=2^{27}\cdot 29^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $78.70$
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:  $2, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{29} a^{6} - \frac{2}{29} a^{5} - \frac{8}{29} a^{4} + \frac{10}{29} a^{3} + \frac{12}{29} a^{2} + \frac{10}{29} a - \frac{7}{29}$, $\frac{1}{29} a^{7} - \frac{12}{29} a^{5} - \frac{6}{29} a^{4} + \frac{3}{29} a^{3} + \frac{5}{29} a^{2} + \frac{13}{29} a - \frac{14}{29}$, $\frac{1}{58} a^{8} + \frac{14}{29} a^{5} + \frac{23}{58} a^{4} - \frac{10}{29} a^{3} + \frac{6}{29} a^{2} - \frac{5}{29} a - \frac{13}{29}$, $\frac{1}{58} a^{9} + \frac{21}{58} a^{5} - \frac{14}{29} a^{4} + \frac{11}{29} a^{3} + \frac{1}{29} a^{2} - \frac{8}{29} a + \frac{11}{29}$, $\frac{1}{116} a^{10} - \frac{1}{116} a^{8} + \frac{1}{116} a^{6} - \frac{4}{29} a^{5} - \frac{15}{116} a^{4} - \frac{1}{29} a^{3} - \frac{9}{29} a^{2} - \frac{13}{29} a + \frac{25}{58}$, $\frac{1}{199657444108} a^{11} + \frac{441679695}{199657444108} a^{10} + \frac{47570027}{199657444108} a^{9} + \frac{84188817}{10508286532} a^{8} - \frac{319721651}{199657444108} a^{7} + \frac{1314196527}{199657444108} a^{6} + \frac{4534818865}{199657444108} a^{5} - \frac{61529673611}{199657444108} a^{4} - \frac{6638643440}{49914361027} a^{3} + \frac{13856810822}{49914361027} a^{2} - \frac{642108941}{99828722054} a - \frac{38017521233}{99828722054}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$ (assuming GRH)

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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 16373551.602 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_4:C_4$ (as 12T30):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 48
The 10 conjugacy class representatives for $A_4:C_4$
Character table for $A_4:C_4$

Intermediate fields

\(\Q(\sqrt{2}) \), 3.3.6728.1 x3, 6.6.362127872.1

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

Sibling fields

Degree 12 sibling: data not computed
Degree 16 sibling: 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}$

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
$2$2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.11.4$x^{4} + 12 x^{2} + 18$$4$$1$$11$$C_4$$[3, 4]$
$29$29.12.10.3$x^{12} + 232 x^{6} + 22707$$6$$2$$10$$C_3 : C_4$$[\ ]_{6}^{2}$