# Properties

 Label 10.4.3301958349971.1 Degree $10$ Signature $[4, 3]$ Discriminant $-3.302\times 10^{12}$ Root discriminant $17.86$ Ramified prime $14891$ Class number $1$ Class group trivial Galois group $S_5$ (as 10T13)

# Learn more

Show commands: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 3*x^9 - 4*x^8 + 14*x^7 + 10*x^6 - 41*x^5 - 3*x^4 + 58*x^3 - 20*x^2 - 24*x + 16)

gp: K = bnfinit(x^10 - 3*x^9 - 4*x^8 + 14*x^7 + 10*x^6 - 41*x^5 - 3*x^4 + 58*x^3 - 20*x^2 - 24*x + 16, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, -24, -20, 58, -3, -41, 10, 14, -4, -3, 1]);

$$x^{10} - 3 x^{9} - 4 x^{8} + 14 x^{7} + 10 x^{6} - 41 x^{5} - 3 x^{4} + 58 x^{3} - 20 x^{2} - 24 x + 16$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $10$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[4, 3]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$-3301958349971$$$$\medspace = -\,14891^{3}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $17.86$ 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: $14891$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Aut(K/\Q)|$: $1$ This field is not Galois over $\Q$. This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{1592} a^{9} + \frac{129}{1592} a^{8} + \frac{77}{398} a^{7} + \frac{37}{796} a^{6} + \frac{113}{796} a^{5} - \frac{457}{1592} a^{4} + \frac{169}{1592} a^{3} - \frac{359}{796} a^{2} - \frac{9}{199} a + \frac{3}{199}$

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: $6$ 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: $$306.644035797$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{4}\cdot(2\pi)^{3}\cdot 306.644035797 \cdot 1}{2\sqrt{3301958349971}}\approx 0.334871601196$

## Galois group

$S_5$ (as 10T13):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A non-solvable group of order 120 The 7 conjugacy class representatives for $S_5$ Character table for $S_5$

## Intermediate fields

 The extension is primitive: there are no intermediate fields between this field and $\Q$.

## Sibling fields

 Degree 5 sibling: 5.3.14891.1 Degree 6 sibling: 6.0.3301958349971.2 Degree 10 sibling: data not computed Degree 12 sibling: data not computed Degree 15 sibling: data not computed Degree 20 siblings: data not computed Degree 24 sibling: data not computed Degree 30 siblings: data not computed Degree 40 sibling: 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 ${\href{/padicField/2.4.0.1}{4} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }$ ${\href{/padicField/3.5.0.1}{5} }^{2}$ ${\href{/padicField/5.5.0.1}{5} }^{2}$ ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ ${\href{/padicField/11.5.0.1}{5} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ ${\href{/padicField/17.2.0.1}{2} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.5.0.1}{5} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ ${\href{/padicField/31.5.0.1}{5} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ ${\href{/padicField/47.3.0.1}{3} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.3.0.1}{3} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
$14891$Data not computed