# Properties

 Label 10.0.262477979.1 Degree $10$ Signature $[0, 5]$ Discriminant $-262477979$ Root discriminant $6.95$ Ramified primes $251, 1045729$ Class number $1$ Class group trivial Galois group $S_{10}$ (as 10T45)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^10 - x^9 + x^8 - 2*x^6 + x^5 - x^4 + x^2 + 1)

gp: K = bnfinit(x^10 - x^9 + x^8 - 2*x^6 + x^5 - x^4 + x^2 + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 1, 0, -1, 1, -2, 0, 1, -1, 1]);

$$x^{10} - x^{9} + x^{8} - 2 x^{6} + x^{5} - x^{4} + x^{2} + 1$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $10$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[0, 5]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$-262477979$$$$\medspace = -\,251\cdot 1045729$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $6.95$ 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: $251, 1045729$ 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$

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: $4$ 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: $$a$$,  $$a^{7} - a^{6} + a^{5} - a^{3}$$,  $$a^{5} - a^{4} - a$$,  $$a^{7} - a^{6} - a^{3} + 1$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$0.6577090020502393$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{5}\cdot 0.6577090020502393 \cdot 1}{2\sqrt{262477979}}\approx 0.198772681025668$

## Galois group

$S_{10}$ (as 10T45):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A non-solvable group of order 3628800 The 42 conjugacy class representatives for $S_{10}$ Character table for $S_{10}$ is not computed

## Intermediate fields

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

## Sibling fields

 Degree 20 sibling: data not computed Degree 45 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 ${\href{/padicField/2.10.0.1}{10} }$ ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.4.0.1}{4} }$ ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.4.0.1}{4} }$ ${\href{/padicField/7.10.0.1}{10} }$ ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ ${\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ ${\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.9.0.1}{9} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.4.0.1}{4} }$ ${\href{/padicField/41.10.0.1}{10} }$ ${\href{/padicField/43.9.0.1}{9} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ ${\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
251Data not computed
1045729Data not computed