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

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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)

Normalized defining 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

10T39:

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

5.5.12398057.1

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

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ 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$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
$23$23.2.1.1$x^{2} - 23$$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