# Properties

 Label 10.0.2379305221151479271.1 Degree 10 Signature $[0, 5]$ Discriminant $-\,7^{2}\cdot 23\cdot 673\cdot 1771151^{2}$ Ramified primes $7, 23, 673, 1771151$ Class number 164 (GRH) Class group [164] (GRH) Galois Group 10T39

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: K<a> := NumberField(PolynomialRing(Rationals())![243, -81, 162, -45, 39, -13, 13, -5, 6, -1, 1]);
sage: K = NumberField(x^10 - x^9 + 6*x^8 - 5*x^7 + 13*x^6 - 13*x^5 + 39*x^4 - 45*x^3 + 162*x^2 - 81*x + 243,"a")
gp: K = bnfinit(x^10 - x^9 + 6*x^8 - 5*x^7 + 13*x^6 - 13*x^5 + 39*x^4 - 45*x^3 + 162*x^2 - 81*x + 243, 1)

## Normalizeddefining polynomial

$x^{10}$ $\mathstrut -\mathstrut x^{9}$ $\mathstrut +\mathstrut 6 x^{8}$ $\mathstrut -\mathstrut 5 x^{7}$ $\mathstrut +\mathstrut 13 x^{6}$ $\mathstrut -\mathstrut 13 x^{5}$ $\mathstrut +\mathstrut 39 x^{4}$ $\mathstrut -\mathstrut 45 x^{3}$ $\mathstrut +\mathstrut 162 x^{2}$ $\mathstrut -\mathstrut 81 x$ $\mathstrut +\mathstrut 243$

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: $-2379305221151479271=-\,7^{2}\cdot 23\cdot 673\cdot 1771151^{2}$ magma: Discriminant(K); sage: K.disc() gp: K.disc Ramified primes: $7, 23, 673, 1771151$ magma: PrimeDivisors(Discriminant(K)); sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ This field is not Galois over $\Q$.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{7} - \frac{1}{9} a^{6} - \frac{1}{3} a^{5} + \frac{4}{9} a^{4} + \frac{4}{9} a^{3} - \frac{4}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{27} a^{8} - \frac{1}{27} a^{7} - \frac{1}{9} a^{6} + \frac{4}{27} a^{5} + \frac{13}{27} a^{4} + \frac{5}{27} a^{3} + \frac{1}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{81} a^{9} - \frac{1}{81} a^{8} - \frac{1}{27} a^{7} + \frac{4}{81} a^{6} + \frac{40}{81} a^{5} + \frac{32}{81} a^{4} + \frac{1}{27} a^{3} - \frac{1}{9} a^{2} - \frac{1}{3} a$

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

## Class group and class number

Multiplicative Abelian group isomorphic to C164, order 164 (assuming GRH)

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: $-1$ magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); sage: UK.torsion_generator() gp: K.tu[2] Fundamental units: $\frac{1}{81} a^{9} - \frac{1}{81} a^{8} + \frac{2}{27} a^{7} - \frac{5}{81} a^{6} + \frac{13}{81} a^{5} - \frac{13}{81} a^{4} + \frac{13}{27} a^{3} - \frac{5}{9} a^{2} + a - 1$,  $\frac{1}{81} a^{9} - \frac{1}{81} a^{8} + \frac{2}{27} a^{7} - \frac{5}{81} a^{6} + \frac{13}{81} a^{5} - \frac{13}{81} a^{4} + \frac{13}{27} a^{3} - \frac{5}{9} a^{2} + a$,  $\frac{1}{27} a^{9} + \frac{2}{9} a^{8} - \frac{1}{27} a^{7} + \frac{19}{27} a^{6} - \frac{4}{27} a^{5} + \frac{8}{9} a^{4} - \frac{16}{27} a^{3} + \frac{43}{9} a^{2} - 3 a + 18$,  $\frac{31}{81} a^{9} - \frac{124}{81} a^{8} + \frac{25}{9} a^{7} - \frac{362}{81} a^{6} + \frac{247}{81} a^{5} - \frac{451}{81} a^{4} + \frac{536}{27} a^{3} - \frac{140}{3} a^{2} + 69 a - 71$ (assuming GRH) magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $1691.97896036$ (assuming GRH) magma: Regulator(K); sage: K.regulator() gp: K.reg

## Galois group

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
 A non-solvable group of order 3840 Conjugacy class representatives for 10T39 Character table for 10T39

## 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

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 ${\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$$\Q_{7}$$x + 2$$1$$1$$0Trivial[\ ] \Q_{7}$$x + 2$$1$$1$$0Trivial[\ ] 7.4.0.1x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$V_4$$[\ ]_{2}^{2} 2323.2.1.1x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.8.0.1$x^{8} + x^{2} - 2 x + 5$$1$$8$$0$$C_8$$[\ ]^{8}$
673Data not computed
1771151Data not computed