# Properties

 Label 15.5.10496055636998343.1 Degree 15 Signature $[5, 5]$ Discriminant $-\,3^{5}\cdot 157\cdot 2377\cdot 115742009$ Ramified primes $3, 157, 2377, 115742009$ 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, 3, -1, -5, 9, -3, -12, 19, -9, -10, 18, -11, -1, 6, -4, 1]);
sage: K = NumberField(x^15 - 4*x^14 + 6*x^13 - x^12 - 11*x^11 + 18*x^10 - 10*x^9 - 9*x^8 + 19*x^7 - 12*x^6 - 3*x^5 + 9*x^4 - 5*x^3 - x^2 + 3*x - 1,"a")
gp: K = bnfinit(x^15 - 4*x^14 + 6*x^13 - x^12 - 11*x^11 + 18*x^10 - 10*x^9 - 9*x^8 + 19*x^7 - 12*x^6 - 3*x^5 + 9*x^4 - 5*x^3 - x^2 + 3*x - 1, 1)

## Normalizeddefining polynomial

$x^{15}$ $\mathstrut -\mathstrut 4 x^{14}$ $\mathstrut +\mathstrut 6 x^{13}$ $\mathstrut -\mathstrut x^{12}$ $\mathstrut -\mathstrut 11 x^{11}$ $\mathstrut +\mathstrut 18 x^{10}$ $\mathstrut -\mathstrut 10 x^{9}$ $\mathstrut -\mathstrut 9 x^{8}$ $\mathstrut +\mathstrut 19 x^{7}$ $\mathstrut -\mathstrut 12 x^{6}$ $\mathstrut -\mathstrut 3 x^{5}$ $\mathstrut +\mathstrut 9 x^{4}$ $\mathstrut -\mathstrut 5 x^{3}$ $\mathstrut -\mathstrut x^{2}$ $\mathstrut +\mathstrut 3 x$ $\mathstrut -\mathstrut 1$

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

## Invariants

 Degree: $15$ magma: Degree(K); sage: K.degree() gp: poldegree(K.pol) Signature: $[5, 5]$ magma: Signature(K); sage: K.signature() gp: K.sign Discriminant: $-10496055636998343=-\,3^{5}\cdot 157\cdot 2377\cdot 115742009$ magma: Discriminant(K); sage: K.disc() gp: K.disc Ramified primes: $3, 157, 2377, 115742009$ 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: $9$ 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: $a$,  $a^{14} - 4 a^{13} + 6 a^{12} - a^{11} - 10 a^{10} + 15 a^{9} - 7 a^{8} - 9 a^{7} + 15 a^{6} - 7 a^{5} - 6 a^{4} + 8 a^{3} - 3 a^{2} - 2 a + 3$,  $a^{14} - 3 a^{13} + 2 a^{12} + 5 a^{11} - 12 a^{10} + 8 a^{9} + 5 a^{8} - 16 a^{7} + 11 a^{6} + a^{5} - 10 a^{4} + 6 a^{3} - a^{2} - 3 a + 2$,  $a^{12} - 4 a^{11} + 7 a^{10} - 5 a^{9} - 4 a^{8} + 12 a^{7} - 11 a^{6} + a^{5} + 5 a^{4} - 5 a^{3} - a^{2} + 2 a - 1$,  $a^{13} - 4 a^{12} + 6 a^{11} - 2 a^{10} - 7 a^{9} + 12 a^{8} - 8 a^{7} - 2 a^{6} + 7 a^{5} - 5 a^{4} + a^{3} + a^{2} - a + 1$,  $a^{14} - 4 a^{13} + 6 a^{12} - 2 a^{11} - 7 a^{10} + 11 a^{9} - 5 a^{8} - 5 a^{7} + 7 a^{6} - a^{5} - 4 a^{4} + 4 a^{3} - a + 1$,  $a^{10} - 3 a^{9} + 3 a^{8} + a^{7} - 6 a^{6} + 5 a^{5} - 5 a^{3} + 2 a^{2} + a - 2$,  $a^{14} - 3 a^{13} + 2 a^{12} + 6 a^{11} - 15 a^{10} + 11 a^{9} + 5 a^{8} - 20 a^{7} + 16 a^{6} - 2 a^{5} - 11 a^{4} + 7 a^{3} - a^{2} - 3 a + 3$,  $a^{14} - 3 a^{13} + 4 a^{12} - 2 a^{11} - 4 a^{10} + 10 a^{9} - 11 a^{8} + 9 a^{6} - 11 a^{5} + a^{4} + 4 a^{3} - 5 a^{2} + 2$ magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $99.2990074596$ 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 $15$ R ${\href{/LocalNumberField/5.11.0.1}{11} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.9.0.1}{9} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ $15$ $15$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.9.0.1}{9} }{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }$ ${\href{/LocalNumberField/41.11.0.1}{11} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{3}$ $15$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }$

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
$3$3.5.0.1$x^{5} - x + 1$$1$$5$$0$$C_5$$[\ ]^{5} 3.10.5.1x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$157$157.2.1.2$x^{2} + 785$$2$$1$$1$$C_2$$[\ ]_{2} 157.13.0.1x^{13} - x + 6$$1$$13$$0$$C_{13}$$[\ ]^{13}$
2377Data not computed
115742009Data not computed