Properties

Label 15.1.24118280788986467.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,59^{6}\cdot 83^{3}$
Root discriminant $12.36$
Ramified primes $59, 83$
Class number $1$
Class group Trivial
Galois Group $S_5 \times S_3$ (as 15T29)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 1, -7, 13, -18, 29, -41, 40, -21, -1, 13, -13, 6, 0, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 + 6*x^12 - 13*x^11 + 13*x^10 - x^9 - 21*x^8 + 40*x^7 - 41*x^6 + 29*x^5 - 18*x^4 + 13*x^3 - 7*x^2 + x + 1)
gp: K = bnfinit(x^15 - 2*x^14 + 6*x^12 - 13*x^11 + 13*x^10 - x^9 - 21*x^8 + 40*x^7 - 41*x^6 + 29*x^5 - 18*x^4 + 13*x^3 - 7*x^2 + x + 1, 1)

Normalized defining polynomial

\(x^{15} \) \(\mathstrut -\mathstrut 2 x^{14} \) \(\mathstrut +\mathstrut 6 x^{12} \) \(\mathstrut -\mathstrut 13 x^{11} \) \(\mathstrut +\mathstrut 13 x^{10} \) \(\mathstrut -\mathstrut x^{9} \) \(\mathstrut -\mathstrut 21 x^{8} \) \(\mathstrut +\mathstrut 40 x^{7} \) \(\mathstrut -\mathstrut 41 x^{6} \) \(\mathstrut +\mathstrut 29 x^{5} \) \(\mathstrut -\mathstrut 18 x^{4} \) \(\mathstrut +\mathstrut 13 x^{3} \) \(\mathstrut -\mathstrut 7 x^{2} \) \(\mathstrut +\mathstrut x \) \(\mathstrut +\mathstrut 1 \)

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:  \(-24118280788986467=-\,59^{6}\cdot 83^{3}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $12.36$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $59, 83$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{28613} a^{14} - \frac{3491}{28613} a^{13} - \frac{9039}{28613} a^{12} + \frac{427}{2201} a^{11} + \frac{273}{2201} a^{10} + \frac{537}{2201} a^{9} - \frac{7047}{28613} a^{8} + \frac{8395}{28613} a^{7} + \frac{9597}{28613} a^{6} - \frac{6764}{28613} a^{5} - \frac{6100}{28613} a^{4} - \frac{5190}{28613} a^{3} - \frac{4106}{28613} a^{2} - \frac{9286}{28613} a + \frac{8939}{28613}$

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:  \( \frac{5066}{28613} a^{14} - \frac{2572}{28613} a^{13} - \frac{10774}{28613} a^{12} + \frac{1800}{2201} a^{11} - \frac{1411}{2201} a^{10} + \frac{6}{2201} a^{9} + \frac{37535}{28613} a^{8} - \frac{75687}{28613} a^{7} + \frac{62141}{28613} a^{6} - \frac{16663}{28613} a^{5} - \frac{560}{28613} a^{4} + \frac{2807}{28613} a^{3} - \frac{27958}{28613} a^{2} + \frac{54122}{28613} a - \frac{9405}{28613} \),  \( \frac{237}{403} a^{14} - \frac{411}{403} a^{13} - \frac{298}{403} a^{12} + \frac{108}{31} a^{11} - \frac{182}{31} a^{10} + \frac{138}{31} a^{9} + \frac{1102}{403} a^{8} - \frac{4832}{403} a^{7} + \frac{7211}{403} a^{6} - \frac{5576}{403} a^{5} + \frac{3085}{403} a^{4} - \frac{2089}{403} a^{3} + \frac{1332}{403} a^{2} - \frac{402}{403} a - \frac{28}{403} \),  \( \frac{28}{403} a^{14} + \frac{181}{403} a^{13} - \frac{411}{403} a^{12} - \frac{10}{31} a^{11} + \frac{80}{31} a^{10} - \frac{154}{31} a^{9} + \frac{1766}{403} a^{8} + \frac{514}{403} a^{7} - \frac{3712}{403} a^{6} + \frac{6063}{403} a^{5} - \frac{4764}{403} a^{4} + \frac{2581}{403} a^{3} - \frac{1725}{403} a^{2} + \frac{1136}{403} a - \frac{374}{403} \),  \( \frac{4962}{28613} a^{14} + \frac{17136}{28613} a^{13} - \frac{14947}{28613} a^{12} - \frac{789}{2201} a^{11} + \frac{3212}{2201} a^{10} - \frac{7420}{2201} a^{9} + \frac{55098}{28613} a^{8} + \frac{24075}{28613} a^{7} - \frac{163396}{28613} a^{6} + \frac{228985}{28613} a^{5} - \frac{138711}{28613} a^{4} + \frac{141985}{28613} a^{3} - \frac{87355}{28613} a^{2} + \frac{75637}{28613} a + \frac{5168}{28613} \),  \( \frac{14906}{28613} a^{14} - \frac{18412}{28613} a^{13} - \frac{25330}{28613} a^{12} + \frac{6173}{2201} a^{11} - \frac{9115}{2201} a^{10} + \frac{3887}{2201} a^{9} + \frac{110193}{28613} a^{8} - \frac{274909}{28613} a^{7} + \frac{331238}{28613} a^{6} - \frac{163650}{28613} a^{5} + \frac{62740}{28613} a^{4} - \frac{49814}{28613} a^{3} + \frac{56397}{28613} a^{2} - \frac{16035}{28613} a - \frac{34620}{28613} \),  \( \frac{5238}{28613} a^{14} - \frac{2151}{28613} a^{13} - \frac{20380}{28613} a^{12} + \frac{2611}{2201} a^{11} - \frac{676}{2201} a^{10} - \frac{4474}{2201} a^{9} + \frac{113036}{28613} a^{8} - \frac{119623}{28613} a^{7} - \frac{3955}{28613} a^{6} + \frac{164740}{28613} a^{5} - \frac{219983}{28613} a^{4} + \frac{140195}{28613} a^{3} - \frac{76091}{28613} a^{2} + \frac{87871}{28613} a - \frac{45612}{28613} \),  \( \frac{24996}{28613} a^{14} - \frac{19999}{28613} a^{13} - \frac{39209}{28613} a^{12} + \frac{9447}{2201} a^{11} - \frac{12398}{2201} a^{10} + \frac{3355}{2201} a^{9} + \frac{166494}{28613} a^{8} - \frac{406904}{28613} a^{7} + \frac{424415}{28613} a^{6} - \frac{199018}{28613} a^{5} + \frac{88916}{28613} a^{4} - \frac{83737}{28613} a^{3} + \frac{115707}{28613} a^{2} + \frac{24413}{28613} a - \frac{28286}{28613} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 81.1679989991 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$S_5 \times S_3$ (as 15T29):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 720
The 21 conjugacy class representatives for $S_5 \times S_3$
Character table for $S_5 \times S_3$ is not computed

Intermediate fields

3.1.59.1, 5.1.4897.1

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

Sibling fields

Degree 18 sibling: data not computed
Degree 30 siblings: data not computed
Degree 36 siblings: data not computed
Degree 45 sibling: 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 ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }$ $15$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ R

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
59Data not computed
83Data not computed