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Properties

Label 12.4.173911685539840000.1
Degree $12$
Signature $[4, 4]$
Discriminant $2^{14}\cdot 5^{4}\cdot 19^{8}$
Root discriminant $27.33$
Ramified primes $2, 5, 19$
Class number $3$
Class group $[3]$
Galois group $A_4\wr C_2$ (as 12T128)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-106, 120, 314, -394, -416, 400, 216, -192, -26, 48, -6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 - 6*x^10 + 48*x^9 - 26*x^8 - 192*x^7 + 216*x^6 + 400*x^5 - 416*x^4 - 394*x^3 + 314*x^2 + 120*x - 106)
 
gp: K = bnfinit(x^12 - 4*x^11 - 6*x^10 + 48*x^9 - 26*x^8 - 192*x^7 + 216*x^6 + 400*x^5 - 416*x^4 - 394*x^3 + 314*x^2 + 120*x - 106, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} - 6 x^{10} + 48 x^{9} - 26 x^{8} - 192 x^{7} + 216 x^{6} + 400 x^{5} - 416 x^{4} - 394 x^{3} + 314 x^{2} + 120 x - 106 \)

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:  \(173911685539840000=2^{14}\cdot 5^{4}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.33$
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:  $2, 5, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{346183} a^{11} - \frac{1332}{3361} a^{10} - \frac{122267}{346183} a^{9} + \frac{103230}{346183} a^{8} + \frac{16344}{346183} a^{7} - \frac{38949}{346183} a^{6} + \frac{156819}{346183} a^{5} - \frac{76947}{346183} a^{4} + \frac{8006}{346183} a^{3} + \frac{79113}{346183} a^{2} - \frac{140966}{346183} a - \frac{105703}{346183}$

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

Class group and class number

$C_{3}$, which has order $3$

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:  \( 11625.7878283 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_4\wr C_2$ (as 12T128):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 288
The 14 conjugacy class representatives for $A_4\wr C_2$
Character table for $A_4\wr C_2$

Intermediate fields

3.3.7220.1

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

Sibling fields

Degree 8 sibling: data not computed
Degree 12 siblings: data not computed
Degree 16 sibling: data not computed
Degree 18 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 sibling: data not computed
Degree 36 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 ${\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ R ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.12.14.3$x^{12} + 2 x^{3} + 2 x^{2} + 2$$12$$1$$14$12T128$[4/3, 4/3, 4/3, 4/3]_{3}^{6}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$19$19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.9.6.2$x^{9} + 228 x^{6} + 16967 x^{3} + 438976$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$