# Properties

 Label 14.14.152...728.1 Degree $14$ Signature $[14, 0]$ Discriminant $1.524\times 10^{20}$ Root discriminant $27.65$ Ramified primes $2, 4129$ Class number $1$ Class group trivial Galois group 14T51

# Related objects

Show commands: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 18*x^12 + 114*x^10 - 320*x^8 + 425*x^6 - 266*x^4 + 76*x^2 - 8)

gp: K = bnfinit(x^14 - 18*x^12 + 114*x^10 - 320*x^8 + 425*x^6 - 266*x^4 + 76*x^2 - 8, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8, 0, 76, 0, -266, 0, 425, 0, -320, 0, 114, 0, -18, 0, 1]);

$$x^{14} - 18 x^{12} + 114 x^{10} - 320 x^{8} + 425 x^{6} - 266 x^{4} + 76 x^{2} - 8$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $14$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[14, 0]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$152387536786230345728$$$$\medspace = 2^{19}\cdot 4129^{4}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $27.65$ 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: $2, 4129$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Aut(K/\Q)|$: $2$ This field is not Galois over $\Q$. This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{7} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{4}$, $\frac{1}{8} a^{13} + \frac{3}{8} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

## Unit group

sage: UK = K.unit_group()

magma: UK, f := UnitGroup(K);

 Rank: $13$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K); Torsion generator: $$-1$$ (order $2$) 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: $$581975.493503$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{14}\cdot(2\pi)^{0}\cdot 581975.493503 \cdot 1}{2\sqrt{152387536786230345728}}\approx 0.386206800122$

## Galois group

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A non-solvable group of order 21504 The 48 conjugacy class representatives for [2^7]L(7)=2wrL(7) Character table for [2^7]L(7)=2wrL(7) is not computed

## Intermediate fields

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

## Sibling fields

 Degree 14 sibling: data not computed Degree 28 siblings: data not computed Degree 42 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 ${\href{/padicField/3.14.0.1}{14} }$ ${\href{/padicField/5.14.0.1}{14} }$ ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ ${\href{/padicField/11.3.0.1}{3} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }$ ${\href{/padicField/13.14.0.1}{14} }$ ${\href{/padicField/17.7.0.1}{7} }^{2}$ ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.7.0.1}{7} }^{2}$ ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ ${\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.7.0.1}{7} }^{2}$ ${\href{/padicField/43.14.0.1}{14} }$ ${\href{/padicField/47.7.0.1}{7} }^{2}$ ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }$

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$2.6.11.3$x^{6} + 2 x^{2} + 14$$6$$1$$11$$S_4\times C_2$$[8/3, 8/3, 3]_{3}^{2} 2.8.8.11x^{8} + 20 x^{2} + 4$$4$$2$$8$$S_4$$[4/3, 4/3]_{3}^{2}$
$4129$Data not computed