# Properties

 Label 12.0.38806720086016.3 Degree $12$ Signature $[0, 6]$ Discriminant $3.881\times 10^{13}$ Root discriminant $13.56$ Ramified primes $2, 23$ Class number $2$ Class group $[2]$ Galois group $C_2 \times S_4$ (as 12T24)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 2*x^10 + 9*x^8 - 24*x^7 + 30*x^6 - 4*x^5 + 6*x^4 - 46*x^3 + 98*x^2 - 70*x + 25)

gp: K = bnfinit(x^12 - 2*x^11 + 2*x^10 + 9*x^8 - 24*x^7 + 30*x^6 - 4*x^5 + 6*x^4 - 46*x^3 + 98*x^2 - 70*x + 25, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, -70, 98, -46, 6, -4, 30, -24, 9, 0, 2, -2, 1]);

$$x^{12} - 2 x^{11} + 2 x^{10} + 9 x^{8} - 24 x^{7} + 30 x^{6} - 4 x^{5} + 6 x^{4} - 46 x^{3} + 98 x^{2} - 70 x + 25$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $12$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[0, 6]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$38806720086016$$$$\medspace = 2^{18}\cdot 23^{6}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $13.56$ sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol))  magma: Abs(Discriminant(Integers(K)))^(1/Degree(K)); Ramified primes: $2, 23$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Aut(K/\Q)|$: $4$ This field is not Galois over $\Q$. This is not a CM field.

## Integral basis (with respect to field generator $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{23} a^{10} + \frac{8}{23} a^{9} + \frac{9}{23} a^{8} - \frac{11}{23} a^{7} + \frac{1}{23} a^{6} + \frac{7}{23} a^{5} + \frac{4}{23} a^{4} + \frac{8}{23} a^{3} + \frac{1}{23} a^{2} + \frac{1}{23} a - \frac{11}{23}$, $\frac{1}{191705} a^{11} - \frac{1197}{191705} a^{10} + \frac{71812}{191705} a^{9} - \frac{564}{1667} a^{8} - \frac{91141}{191705} a^{7} + \frac{16696}{191705} a^{6} - \frac{6208}{38341} a^{5} - \frac{22959}{191705} a^{4} - \frac{44484}{191705} a^{3} - \frac{77311}{191705} a^{2} - \frac{31737}{191705} a - \frac{9723}{38341}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

$C_{2}$, which has order $2$

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

## Unit group

sage: UK = K.unit_group()

magma: UK, f := UnitGroup(K);

 Rank: $5$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K); Torsion generator: $$-\frac{921}{8335} a^{11} + \frac{2217}{8335} a^{10} - \frac{627}{8335} a^{9} - \frac{177}{1667} a^{8} - \frac{9259}{8335} a^{7} + \frac{26064}{8335} a^{6} - \frac{3576}{1667} a^{5} - \frac{8991}{8335} a^{4} - \frac{5096}{8335} a^{3} + \frac{64206}{8335} a^{2} - \frac{76083}{8335} a + \frac{6427}{1667}$$ (order $4$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); 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 sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$103.136133603$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{6}\cdot 103.136133603 \cdot 2}{4\sqrt{38806720086016}}\approx 0.509338601920$

## Galois group

$C_2\times S_4$ (as 12T24):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A solvable group of order 48 The 10 conjugacy class representatives for $C_2 \times S_4$ Character table for $C_2 \times S_4$

## Intermediate fields

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

## Sibling fields

 Degree 6 siblings: data not computed Degree 8 siblings: data not computed Degree 12 siblings: 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])

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];

## 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$$[2, 2, 2]^{3} 2323.4.2.1x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2} 23.4.2.1x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$