↑

Properties

Label 12.0.972102421841920000.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{14}\cdot 5^{4}\cdot 37^{7}$
Root discriminant $31.55$
Ramified primes $2, 5, 37$
Class number $4$
Class group $[2, 2]$
Galois group $A_4:C_4$ (as 12T27)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![27, -144, 360, -488, 311, 14, -124, 32, 37, 0, 2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 2*x^10 + 37*x^8 + 32*x^7 - 124*x^6 + 14*x^5 + 311*x^4 - 488*x^3 + 360*x^2 - 144*x + 27)
 
gp: K = bnfinit(x^12 - 2*x^11 + 2*x^10 + 37*x^8 + 32*x^7 - 124*x^6 + 14*x^5 + 311*x^4 - 488*x^3 + 360*x^2 - 144*x + 27, 1)
 

Normalized defining polynomial

\( x^{12} - 2 x^{11} + 2 x^{10} + 37 x^{8} + 32 x^{7} - 124 x^{6} + 14 x^{5} + 311 x^{4} - 488 x^{3} + 360 x^{2} - 144 x + 27 \)

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:  \(972102421841920000=2^{14}\cdot 5^{4}\cdot 37^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.55$
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:  $2, 5, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{111} a^{10} + \frac{5}{37} a^{9} + \frac{3}{37} a^{8} - \frac{5}{37} a^{7} + \frac{29}{111} a^{6} + \frac{9}{37} a^{5} - \frac{49}{111} a^{4} + \frac{11}{37} a^{3} + \frac{1}{3} a^{2} - \frac{17}{37} a + \frac{16}{37}$, $\frac{1}{27078561} a^{11} - \frac{22598}{27078561} a^{10} - \frac{2892445}{27078561} a^{9} - \frac{483129}{3008729} a^{8} + \frac{756547}{27078561} a^{7} - \frac{4948675}{27078561} a^{6} - \frac{3950812}{27078561} a^{5} - \frac{6199504}{27078561} a^{4} + \frac{12939368}{27078561} a^{3} + \frac{3412459}{27078561} a^{2} + \frac{3939508}{9026187} a + \frac{199028}{3008729}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$

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:  \( -1 \) (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:  \( 4595.70576439 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_4:C_4$ (as 12T27):

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

3.3.148.1, 6.6.81044800.1

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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}$

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$2.12.14.2$x^{12} + 2 x^{4} + 2 x^{3} + 2$$12$$1$$14$12T27$[4/3, 4/3]_{3}^{4}$
$5$5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$37$$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.4.3.1$x^{4} - 37$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.3.1$x^{4} - 37$$4$$1$$3$$C_4$$[\ ]_{4}$