# Properties

 Label 10.0.246071287.1 Degree $10$ Signature $[0, 5]$ Discriminant $-\,7^{5}\cdot 11^{4}$ Root discriminant $6.90$ Ramified primes $7, 11$ Class number $1$ Class group Trivial Galois Group $D_5\times C_5$ (as 10T6)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 4, -7, 12, -15, 15, -12, 7, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 3*x^9 + 7*x^8 - 12*x^7 + 15*x^6 - 15*x^5 + 12*x^4 - 7*x^3 + 4*x^2 - 2*x + 1)
gp: K = bnfinit(x^10 - 3*x^9 + 7*x^8 - 12*x^7 + 15*x^6 - 15*x^5 + 12*x^4 - 7*x^3 + 4*x^2 - 2*x + 1, 1)

## Normalizeddefining polynomial

$$x^{10}$$ $$\mathstrut -\mathstrut 3 x^{9}$$ $$\mathstrut +\mathstrut 7 x^{8}$$ $$\mathstrut -\mathstrut 12 x^{7}$$ $$\mathstrut +\mathstrut 15 x^{6}$$ $$\mathstrut -\mathstrut 15 x^{5}$$ $$\mathstrut +\mathstrut 12 x^{4}$$ $$\mathstrut -\mathstrut 7 x^{3}$$ $$\mathstrut +\mathstrut 4 x^{2}$$ $$\mathstrut -\mathstrut 2 x$$ $$\mathstrut +\mathstrut 1$$

magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol

## Invariants

 Degree: $10$ magma: Degree(K); sage: K.degree() gp: poldegree(K.pol) Signature: $[0, 5]$ magma: Signature(K); sage: K.signature() gp: K.sign Discriminant: $$-246071287=-\,7^{5}\cdot 11^{4}$$ magma: Discriminant(K); sage: K.disc() gp: K.disc Root discriminant: $6.90$ magma: Abs(Discriminant(K))^(1/Degree(K)); sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $7, 11$ magma: PrimeDivisors(Discriminant(K)); sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ This field is not Galois over $\Q$. This is not a CM field.

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

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

magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk

## Class group and class number

Trivial Abelian group, order $1$

magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp

## Unit group

magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
 Rank: $4$ magma: UnitRank(K); sage: UK.rank() gp: K.fu Torsion generator: $$-1$$ (order $2$) magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); sage: UK.torsion_generator() gp: K.tu[2] Fundamental units: $$a^{9} - 3 a^{8} + 6 a^{7} - 9 a^{6} + 9 a^{5} - 6 a^{4} + 3 a^{3}$$,  $$a^{9} - 3 a^{8} + 7 a^{7} - 12 a^{6} + 14 a^{5} - 14 a^{4} + 10 a^{3} - 5 a^{2} + 3 a - 1$$,  $$a^{8} - 2 a^{7} + 4 a^{6} - 5 a^{5} + 4 a^{4} - 2 a^{3} + a^{2} + 1$$,  $$a^{9} - 2 a^{8} + 4 a^{7} - 5 a^{6} + 4 a^{5} - 2 a^{4} + a^{2}$$ magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $$0.634803768223$$ magma: Regulator(K); sage: K.regulator() gp: K.reg

## Galois group

$C_5\times D_5$ (as 10T6):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
 A solvable group of order 50 The 20 conjugacy class representatives for $D_5\times C_5$ Character table for $D_5\times C_5$

## 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 25 sibling: data not computed

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 ${\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }$ ${\href{/LocalNumberField/5.10.0.1}{10} }$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/59.10.0.1}{10} }$

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

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];
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]])

## Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$7$7.10.5.2$x^{10} - 2401 x^{2} + 67228$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5} 1111.5.0.1x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$