# Properties

 Label 15.3.2354315214908549.1 Degree $15$ Signature $[3, 6]$ Discriminant $27765763\cdot 84792023$ Root discriminant $10.59$ Ramified primes $27765763, 84792023$ Class number $1$ Class group Trivial Galois Group 15T104

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 2, -1, 2, 0, 0, -1, -2, 2, -2, 3, -1, -1, 1, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 + x^13 - x^12 - x^11 + 3*x^10 - 2*x^9 + 2*x^8 - 2*x^7 - x^6 + 2*x^3 - x^2 + 2*x - 1)
gp: K = bnfinit(x^15 - x^14 + x^13 - x^12 - x^11 + 3*x^10 - 2*x^9 + 2*x^8 - 2*x^7 - x^6 + 2*x^3 - x^2 + 2*x - 1, 1)

## Normalizeddefining polynomial

$$x^{15}$$ $$\mathstrut -\mathstrut x^{14}$$ $$\mathstrut +\mathstrut x^{13}$$ $$\mathstrut -\mathstrut x^{12}$$ $$\mathstrut -\mathstrut x^{11}$$ $$\mathstrut +\mathstrut 3 x^{10}$$ $$\mathstrut -\mathstrut 2 x^{9}$$ $$\mathstrut +\mathstrut 2 x^{8}$$ $$\mathstrut -\mathstrut 2 x^{7}$$ $$\mathstrut -\mathstrut x^{6}$$ $$\mathstrut +\mathstrut 2 x^{3}$$ $$\mathstrut -\mathstrut x^{2}$$ $$\mathstrut +\mathstrut 2 x$$ $$\mathstrut -\mathstrut 1$$

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

## Invariants

 Degree: $15$ magma: Degree(K); sage: K.degree() gp: poldegree(K.pol) Signature: $[3, 6]$ magma: Signature(K); sage: K.signature() gp: K.sign Discriminant: $$2354315214908549=27765763\cdot 84792023$$ magma: Discriminant(K); sage: K.disc() gp: K.disc Root discriminant: $10.59$ magma: Abs(Discriminant(K))^(1/Degree(K)); sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $27765763, 84792023$ 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$

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: $8$ 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: $$a$$,  $$a^{13} - a^{12} + a^{8} - a^{6} - a^{5} + a^{4} - a^{3} + 1$$,  $$a^{14} - 2 a^{13} + 2 a^{12} - a^{11} - a^{10} + 3 a^{9} - 3 a^{8} + 2 a^{7} - a^{6} - a^{4} + 2 a^{2} - a + 1$$,  $$a^{12} - a^{10} + a^{9} - a^{8} + 2 a^{7} + a^{6} - a^{5} - a^{4} - a^{3} + a^{2} - a + 2$$,  $$a^{13} - a^{12} - a^{9} + 2 a^{8} - a^{6} - a^{5} - a^{3} + a^{2} + a$$,  $$a^{14} - 2 a^{13} + a^{11} - a^{10} + 3 a^{9} - 2 a^{8} - 2 a^{7} + 2 a^{5} + a^{4} + a^{2} - 2 a$$,  $$a^{14} - a^{13} + a^{11} - 2 a^{10} + 3 a^{9} - a^{8} - a^{7} - a^{5} - a^{3} + 2 a^{2} - 2 a$$,  $$a^{14} - a^{13} + a^{12} - 2 a^{10} + 3 a^{9} - 2 a^{8} + a^{7} - a^{5} - a^{3} + 2 a^{2} - 2 a + 2$$ magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $$31.1653663197$$ 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 1307674368000 The 176 conjugacy class representatives for S15 are not computed Character table for S15 is not computed

## Intermediate fields

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

## Sibling fields

 Degree 30 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 ${\href{/LocalNumberField/2.9.0.1}{9} }{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }$ ${\href{/LocalNumberField/3.11.0.1}{11} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ $15$ ${\href{/LocalNumberField/7.7.0.1}{7} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.11.0.1}{11} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.9.0.1}{9} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.11.0.1}{11} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.9.0.1}{9} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }$ $15$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$

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
27765763Data not computed
84792023Data not computed