Properties

Label 15.3.2432060458605625.1
Degree $15$
Signature $[3, 6]$
Discriminant $5^{4}\cdot 19^{2}\cdot 47^{6}$
Root discriminant $10.61$
Ramified primes $5, 19, 47$
Class number $1$
Class group Trivial
Galois Group 15T46

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Show commands for: Magma / SageMath / Pari/GP

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

Normalized defining polynomial

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

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:  \(2432060458605625=5^{4}\cdot 19^{2}\cdot 47^{6}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $10.61$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $5, 19, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{14027} a^{14} + \frac{4447}{14027} a^{13} + \frac{168}{1079} a^{12} - \frac{6278}{14027} a^{11} + \frac{3215}{14027} a^{10} + \frac{6803}{14027} a^{9} + \frac{3501}{14027} a^{8} + \frac{2481}{14027} a^{7} - \frac{289}{1079} a^{6} - \frac{4981}{14027} a^{5} - \frac{6858}{14027} a^{4} + \frac{4346}{14027} a^{3} + \frac{139}{1079} a^{2} + \frac{64}{14027} a + \frac{4128}{14027}$

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:  \( \frac{7272}{14027} a^{14} - \frac{7678}{14027} a^{13} - \frac{811}{1079} a^{12} + \frac{4269}{14027} a^{11} + \frac{10498}{14027} a^{10} - \frac{29867}{14027} a^{9} - \frac{27787}{14027} a^{8} + \frac{17137}{14027} a^{7} + \frac{1363}{1079} a^{6} - \frac{18145}{14027} a^{5} - \frac{19418}{14027} a^{4} + \frac{43362}{14027} a^{3} + \frac{1943}{1079} a^{2} + \frac{2517}{14027} a - \frac{12991}{14027} \),  \( \frac{9212}{14027} a^{14} - \frac{7103}{14027} a^{13} - \frac{1828}{1079} a^{12} + \frac{14412}{14027} a^{11} + \frac{19610}{14027} a^{10} - \frac{45481}{14027} a^{9} - \frac{38942}{14027} a^{8} + \frac{33043}{14027} a^{7} + \frac{1783}{1079} a^{6} - \frac{30709}{14027} a^{5} - \frac{26342}{14027} a^{4} + \frac{44375}{14027} a^{3} + \frac{2932}{1079} a^{2} - \frac{13593}{14027} a - \frac{14088}{14027} \),  \( \frac{6638}{14027} a^{14} - \frac{7649}{14027} a^{13} - \frac{502}{1079} a^{12} + \frac{853}{14027} a^{11} + \frac{6103}{14027} a^{10} - \frac{22653}{14027} a^{9} - \frac{17128}{14027} a^{8} + \frac{1180}{14027} a^{7} + \frac{1159}{1079} a^{6} - \frac{16266}{14027} a^{5} - \frac{5789}{14027} a^{4} + \frac{23263}{14027} a^{3} + \frac{1216}{1079} a^{2} + \frac{4022}{14027} a - \frac{7094}{14027} \),  \( \frac{7851}{14027} a^{14} - \frac{13833}{14027} a^{13} - \frac{649}{1079} a^{12} + \frac{16327}{14027} a^{11} + \frac{6392}{14027} a^{10} - \frac{32517}{14027} a^{9} - \frac{6569}{14027} a^{8} + \frac{36909}{14027} a^{7} + \frac{1277}{1079} a^{6} - \frac{26609}{14027} a^{5} - \frac{6532}{14027} a^{4} + \frac{48863}{14027} a^{3} + \frac{420}{1079} a^{2} - \frac{30562}{14027} a - \frac{21496}{14027} \),  \( \frac{2424}{14027} a^{14} - \frac{7235}{14027} a^{13} + \frac{449}{1079} a^{12} + \frac{1423}{14027} a^{11} - \frac{5852}{14027} a^{10} - \frac{5280}{14027} a^{9} + \frac{14116}{14027} a^{8} - \frac{3639}{14027} a^{7} - \frac{265}{1079} a^{6} + \frac{3303}{14027} a^{5} - \frac{1797}{14027} a^{4} + \frac{427}{14027} a^{3} - \frac{791}{1079} a^{2} + \frac{839}{14027} a - \frac{9006}{14027} \),  \( \frac{406}{14027} a^{14} - \frac{4001}{14027} a^{13} + \frac{231}{1079} a^{12} + \frac{4046}{14027} a^{11} + \frac{779}{14027} a^{10} - \frac{1301}{14027} a^{9} + \frac{4679}{14027} a^{8} + \frac{11369}{14027} a^{7} + \frac{277}{1079} a^{6} - \frac{2398}{14027} a^{5} - \frac{7002}{14027} a^{4} + \frac{11101}{14027} a^{3} - \frac{753}{1079} a^{2} - \frac{16097}{14027} a - \frac{21299}{14027} \),  \( \frac{4751}{14027} a^{14} + \frac{3035}{14027} a^{13} - \frac{1371}{1079} a^{12} - \frac{5376}{14027} a^{11} + \frac{13089}{14027} a^{10} - \frac{11182}{14027} a^{9} - \frac{44852}{14027} a^{8} - \frac{9476}{14027} a^{7} + \frac{1607}{1079} a^{6} - \frac{1182}{14027} a^{5} - \frac{25691}{14027} a^{4} + \frac{14129}{14027} a^{3} + \frac{3278}{1079} a^{2} + \frac{9497}{14027} a - \frac{11645}{14027} \),  \( \frac{1426}{14027} a^{14} + \frac{1218}{14027} a^{13} - \frac{1049}{1079} a^{12} + \frac{10825}{14027} a^{11} + \frac{11788}{14027} a^{10} - \frac{19633}{14027} a^{9} - \frac{15213}{14027} a^{8} + \frac{31156}{14027} a^{7} + \frac{1143}{1079} a^{6} - \frac{19271}{14027} a^{5} - \frac{16716}{14027} a^{4} + \frac{25516}{14027} a^{3} + \frac{1836}{1079} a^{2} - \frac{34979}{14027} a - \frac{18839}{14027} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 32.0155843928 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

15T46:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 2430
The 72 conjugacy class representatives for [3^5]D(5)=3wrD(5) are not computed
Character table for [3^5]D(5)=3wrD(5) is not computed

Intermediate fields

5.1.2209.1

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

Sibling fields

Degree 15 siblings: data not computed
Degree 30 siblings: data not computed
Degree 45 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 $15$ $15$ R ${\href{/LocalNumberField/7.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ $15$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ R $15$ $15$

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
$5$5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
5.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.6.0.1$x^{6} - x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
$47$47.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
47.6.3.2$x^{6} - 2209 x^{2} + 207646$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
47.6.3.2$x^{6} - 2209 x^{2} + 207646$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$