# Properties

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

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: K<a> := NumberField(PolynomialRing(Rationals())![-1, 2, -1, 2, 0, 0, -1, -2, 2, -2, 3, -1, -1, 1, -1, 1]);
sage: K = 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,"a")
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 Ramified primes: $27765763, 84792023$ magma: PrimeDivisors(Discriminant(K)); sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ This field is not Galois over $\Q$.

## 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: $-1$ 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 Conjugacy class representatives for 15T104 Character table for 15T104

## Intermediate fields

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

## 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