Properties

Label 15.1.96318312824155136.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{10}\cdot 11^{7}\cdot 13^{6}$
Root discriminant $13.56$
Ramified primes $2, 11, 13$
Class number $1$
Class group Trivial
Galois Group $D_5\times S_3$ (as 15T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 3, -4, 12, -20, 33, -52, 65, -69, 68, -67, 58, -39, 19, -6, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 6*x^14 + 19*x^13 - 39*x^12 + 58*x^11 - 67*x^10 + 68*x^9 - 69*x^8 + 65*x^7 - 52*x^6 + 33*x^5 - 20*x^4 + 12*x^3 - 4*x^2 + 3*x - 1)
gp: K = bnfinit(x^15 - 6*x^14 + 19*x^13 - 39*x^12 + 58*x^11 - 67*x^10 + 68*x^9 - 69*x^8 + 65*x^7 - 52*x^6 + 33*x^5 - 20*x^4 + 12*x^3 - 4*x^2 + 3*x - 1, 1)

Normalized defining polynomial

\(x^{15} \) \(\mathstrut -\mathstrut 6 x^{14} \) \(\mathstrut +\mathstrut 19 x^{13} \) \(\mathstrut -\mathstrut 39 x^{12} \) \(\mathstrut +\mathstrut 58 x^{11} \) \(\mathstrut -\mathstrut 67 x^{10} \) \(\mathstrut +\mathstrut 68 x^{9} \) \(\mathstrut -\mathstrut 69 x^{8} \) \(\mathstrut +\mathstrut 65 x^{7} \) \(\mathstrut -\mathstrut 52 x^{6} \) \(\mathstrut +\mathstrut 33 x^{5} \) \(\mathstrut -\mathstrut 20 x^{4} \) \(\mathstrut +\mathstrut 12 x^{3} \) \(\mathstrut -\mathstrut 4 x^{2} \) \(\mathstrut +\mathstrut 3 x \) \(\mathstrut -\mathstrut 1 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[1, 7]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(-96318312824155136=-\,2^{10}\cdot 11^{7}\cdot 13^{6}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $13.56$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 11, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{11} a^{13} - \frac{2}{11} a^{12} - \frac{4}{11} a^{11} - \frac{3}{11} a^{10} - \frac{4}{11} a^{9} - \frac{5}{11} a^{8} - \frac{2}{11} a^{7} - \frac{2}{11} a^{6} - \frac{1}{11} a^{5} - \frac{4}{11} a^{4} - \frac{1}{11} a^{3} + \frac{3}{11} a^{2} - \frac{5}{11} a - \frac{3}{11}$, $\frac{1}{319} a^{14} - \frac{10}{319} a^{13} + \frac{1}{319} a^{12} + \frac{73}{319} a^{11} - \frac{2}{319} a^{10} + \frac{115}{319} a^{9} + \frac{159}{319} a^{8} - \frac{96}{319} a^{7} - \frac{73}{319} a^{6} + \frac{37}{319} a^{5} - \frac{57}{319} a^{4} + \frac{11}{29} a^{3} - \frac{95}{319} a^{2} - \frac{117}{319} a + \frac{123}{319}$

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:  $7$
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:  \( \frac{163}{319} a^{14} - \frac{615}{319} a^{13} + \frac{1323}{319} a^{12} - \frac{1412}{319} a^{11} + \frac{776}{319} a^{10} + \frac{30}{29} a^{9} + \frac{107}{319} a^{8} - \frac{771}{319} a^{7} + \frac{426}{319} a^{6} + \frac{21}{29} a^{5} - \frac{591}{319} a^{4} - \frac{751}{319} a^{3} + \frac{1}{319} a^{2} - \frac{540}{319} a + \frac{97}{319} \),  \( \frac{195}{319} a^{14} - \frac{848}{319} a^{13} + \frac{2138}{319} a^{12} - \frac{3252}{319} a^{11} + \frac{331}{29} a^{10} - \frac{3037}{319} a^{9} + \frac{3165}{319} a^{8} - \frac{3379}{319} a^{7} + \frac{2382}{319} a^{6} - \frac{1224}{319} a^{5} + \frac{108}{319} a^{4} - \frac{1113}{319} a^{3} + \frac{93}{319} a^{2} + \frac{6}{29} a + \frac{263}{319} \),  \( \frac{33}{29} a^{14} - \frac{1774}{319} a^{13} + \frac{4945}{319} a^{12} - \frac{8678}{319} a^{11} + \frac{11251}{319} a^{10} - \frac{10977}{319} a^{9} + \frac{10476}{319} a^{8} - \frac{10169}{319} a^{7} + \frac{8707}{319} a^{6} - \frac{5970}{319} a^{5} + \frac{2509}{319} a^{4} - \frac{1955}{319} a^{3} - \frac{207}{319} a^{2} - \frac{392}{319} a + \frac{163}{319} \),  \( \frac{13}{319} a^{14} - \frac{43}{319} a^{13} + \frac{158}{319} a^{12} - \frac{356}{319} a^{11} + \frac{670}{319} a^{10} - \frac{767}{319} a^{9} + \frac{675}{319} a^{8} - \frac{146}{319} a^{7} - \frac{166}{319} a^{6} + \frac{394}{319} a^{5} - \frac{70}{29} a^{4} + \frac{529}{319} a^{3} - \frac{655}{319} a^{2} - \frac{42}{319} a - \frac{257}{319} \),  \( \frac{183}{319} a^{14} - \frac{1134}{319} a^{13} + \frac{3576}{319} a^{12} - \frac{7289}{319} a^{11} + \frac{10625}{319} a^{10} - \frac{12044}{319} a^{9} + \frac{11900}{319} a^{8} - \frac{11942}{319} a^{7} + \frac{979}{29} a^{6} - \frac{7961}{319} a^{5} + \frac{4649}{319} a^{4} - \frac{2478}{319} a^{3} + \frac{1291}{319} a^{2} + \frac{310}{319} a + \frac{324}{319} \),  \( a - 1 \),  \( \frac{60}{319} a^{14} - \frac{426}{319} a^{13} + \frac{1307}{319} a^{12} - \frac{2377}{319} a^{11} + \frac{2548}{319} a^{10} - \frac{132}{29} a^{9} + \frac{57}{319} a^{8} - \frac{47}{319} a^{7} + \frac{57}{319} a^{6} + \frac{128}{29} a^{5} - \frac{2521}{319} a^{4} + \frac{1982}{319} a^{3} - \frac{393}{319} a^{2} + \frac{723}{319} a - \frac{479}{319} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 226.323474369 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$S_3\times D_5$ (as 15T7):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 60
The 12 conjugacy class representatives for $D_5\times S_3$
Character table for $D_5\times S_3$

Intermediate fields

3.1.44.1, 5.1.20449.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 30 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 R $15$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ R R ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ $15$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
2Data not computed
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$