# Properties

 Label 10.0.262091507.1 Degree $10$ Signature $[0, 5]$ Discriminant $-262091507$ Root discriminant $6.95$ Ramified primes $83, 1777$ Class number $1$ Class group trivial Galois group $C_2 \wr S_5$ (as 10T39)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^10 - x^9 - 2*x^6 + 3*x^5 + 3*x^4 - 4*x^3 - 2*x^2 + 2*x + 1)

gp: K = bnfinit(x^10 - x^9 - 2*x^6 + 3*x^5 + 3*x^4 - 4*x^3 - 2*x^2 + 2*x + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, -2, -4, 3, 3, -2, 0, 0, -1, 1]);

$$x^{10} - x^{9} - 2 x^{6} + 3 x^{5} + 3 x^{4} - 4 x^{3} - 2 x^{2} + 2 x + 1$$

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: $$-262091507$$$$\medspace = -\,83\cdot 1777^{2}$$ 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: $83, 1777$ 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 $$a$$)

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

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: $$5 a^{9} - 8 a^{8} + 5 a^{7} - 3 a^{6} - 8 a^{5} + 20 a^{4} + 3 a^{3} - 21 a^{2} + 3 a + 8$$,  $$3 a^{9} - 6 a^{8} + 4 a^{7} - 3 a^{6} - 4 a^{5} + 14 a^{4} - a^{3} - 14 a^{2} + 3 a + 5$$,  $$4 a^{9} - 7 a^{8} + 4 a^{7} - 2 a^{6} - 7 a^{5} + 18 a^{4} + a^{3} - 20 a^{2} + 4 a + 7$$,  $$4 a^{9} - 6 a^{8} + 4 a^{7} - 3 a^{6} - 6 a^{5} + 14 a^{4} + 3 a^{3} - 15 a^{2} + 2 a + 5$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$0.656821013577$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{5}\cdot 0.656821013577 \cdot 1}{2\sqrt{262091507}}\approx 0.1986506135661$

## Galois group

$C_2 \wr S_5$ (as 10T39):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A non-solvable group of order 3840 The 36 conjugacy class representatives for $C_2 \wr S_5$ Character table for $C_2 \wr S_5$ 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 10 sibling: data not computed Degree 20 siblings: data not computed Degree 30 siblings: data not computed Degree 32 siblings: data not computed Degree 40 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 ${\href{/padicField/2.10.0.1}{10} }$ ${\href{/padicField/3.5.0.1}{5} }^{2}$ ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ ${\href{/padicField/13.10.0.1}{10} }$ ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.5.0.1}{5} }^{2}$ ${\href{/padicField/37.5.0.1}{5} }^{2}$ ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ ${\href{/padicField/43.10.0.1}{10} }$ ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.10.0.1}{10} }$ ${\href{/padicField/59.5.0.1}{5} }^{2}$

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
$83$83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2} 83.4.0.1x^{4} - x + 22$$1$$4$$0$$C_4$$[\ ]^{4}$
83.4.0.1$x^{4} - x + 22$$1$$4$$0$$C_4$$[\ ]^{4}$
1777Data not computed