Properties

Label 15.3.31123779291971584.1
Degree $15$
Signature $[3, 6]$
Discriminant $2^{18}\cdot 587^{4}$
Root discriminant $12.58$
Ramified primes $2, 587$
Class number $1$
Class group Trivial
Galois Group 15T28

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Show commands for: Magma / SageMath / Pari/GP

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

Normalized defining polynomial

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

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

Invariants

Degree:  $15$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[3, 6]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(31123779291971584=2^{18}\cdot 587^{4}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $12.58$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 587$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3501} a^{14} + \frac{436}{3501} a^{13} + \frac{1043}{3501} a^{12} - \frac{383}{3501} a^{11} + \frac{1055}{3501} a^{10} + \frac{1345}{3501} a^{9} + \frac{1748}{3501} a^{8} - \frac{1091}{3501} a^{7} + \frac{464}{3501} a^{6} - \frac{752}{3501} a^{5} + \frac{1220}{3501} a^{4} - \frac{236}{3501} a^{3} - \frac{1372}{3501} a^{2} + \frac{475}{3501} a + \frac{538}{3501}$

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:  \( a \),  \( \frac{431}{1167} a^{14} - \frac{749}{1167} a^{13} + \frac{1016}{1167} a^{12} - \frac{1304}{1167} a^{11} - \frac{814}{1167} a^{10} + \frac{1252}{1167} a^{9} - \frac{2050}{1167} a^{8} + \frac{1636}{1167} a^{7} + \frac{38}{1167} a^{6} - \frac{1631}{1167} a^{5} + \frac{281}{1167} a^{4} + \frac{1369}{1167} a^{3} - \frac{1219}{1167} a^{2} - \frac{278}{1167} a - \frac{248}{389} \),  \( \frac{248}{389} a^{14} - \frac{431}{1167} a^{13} + \frac{1493}{1167} a^{12} - \frac{1760}{1167} a^{11} - \frac{2416}{1167} a^{10} - \frac{2162}{1167} a^{9} - \frac{4972}{1167} a^{8} + \frac{1306}{1167} a^{7} + \frac{1340}{1167} a^{6} + \frac{1450}{1167} a^{5} + \frac{143}{1167} a^{4} - \frac{3257}{1167} a^{3} - \frac{3922}{1167} a^{2} - \frac{2924}{1167} a - \frac{1954}{1167} \),  \( \frac{1496}{3501} a^{14} - \frac{97}{3501} a^{13} + \frac{49}{3501} a^{12} + \frac{29}{3501} a^{11} - \frac{10007}{3501} a^{10} - \frac{2122}{3501} a^{9} - \frac{2573}{3501} a^{8} - \frac{1837}{3501} a^{7} + \frac{12616}{3501} a^{6} - \frac{2338}{3501} a^{5} - \frac{4736}{3501} a^{4} - \frac{4123}{3501} a^{3} - \frac{3260}{3501} a^{2} - \frac{1270}{3501} a - \frac{2716}{3501} \),  \( \frac{1054}{3501} a^{14} - \frac{254}{3501} a^{13} + \frac{1175}{3501} a^{12} - \frac{2234}{3501} a^{11} - \frac{3682}{3501} a^{10} - \frac{4943}{3501} a^{9} - \frac{1468}{3501} a^{8} + \frac{748}{3501} a^{7} + \frac{3584}{3501} a^{6} + \frac{4453}{3501} a^{5} - \frac{8323}{3501} a^{4} - \frac{1340}{3501} a^{3} - \frac{2509}{3501} a^{2} - \frac{1160}{3501} a - \frac{2444}{3501} \),  \( \frac{322}{3501} a^{14} + \frac{1519}{3501} a^{13} - \frac{1417}{3501} a^{12} + \frac{376}{3501} a^{11} - \frac{4555}{3501} a^{10} - \frac{6869}{3501} a^{9} + \frac{1529}{3501} a^{8} + \frac{3466}{3501} a^{7} + \frac{8201}{3501} a^{6} + \frac{7594}{3501} a^{5} - \frac{7441}{3501} a^{4} - \frac{11807}{3501} a^{3} - \frac{1825}{3501} a^{2} + \frac{3574}{3501} a + \frac{4021}{3501} \),  \( \frac{4682}{3501} a^{14} - \frac{898}{3501} a^{13} + \frac{4099}{3501} a^{12} - \frac{5362}{3501} a^{11} - \frac{23741}{3501} a^{10} - \frac{12679}{3501} a^{9} - \frac{17540}{3501} a^{8} + \frac{2231}{3501} a^{7} + \frac{24001}{3501} a^{6} + \frac{3476}{3501} a^{5} - \frac{14429}{3501} a^{4} - \frac{20809}{3501} a^{3} - \frac{19208}{3501} a^{2} - \frac{10855}{3501} a - \frac{4138}{3501} \),  \( \frac{218}{3501} a^{14} - \frac{646}{3501} a^{13} + \frac{976}{3501} a^{12} - \frac{637}{3501} a^{11} + \frac{91}{3501} a^{10} + \frac{1460}{3501} a^{9} - \frac{2879}{3501} a^{8} - \frac{937}{3501} a^{7} - \frac{2711}{3501} a^{6} - \frac{556}{3501} a^{5} + \frac{4552}{3501} a^{4} + \frac{3401}{3501} a^{3} + \frac{3157}{3501} a^{2} - \frac{2647}{3501} a - \frac{4084}{3501} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 152.23669177 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

15T28:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 720
The 11 conjugacy class representatives for S_6(15)
Character table for S_6(15)

Intermediate fields

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

Sibling fields

Degree 6 siblings: 6.2.37568.1, 6.2.51779072768.2
Degree 10 sibling: 10.2.828465164288.1
Degree 12 siblings: Deg 12, Deg 12
Degree 15 sibling: Deg 15
Degree 20 siblings: Deg 20, 20.4.2745418113754971322187776.1, Deg 20
Degree 30 siblings: data not computed
Degree 36 sibling: data not computed
Degree 40 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 R ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$2$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.12.18.52$x^{12} + 20 x^{11} - 22 x^{10} - 24 x^{9} + 26 x^{8} - 24 x^{7} + 8 x^{6} + 32 x^{5} + 28 x^{4} + 16 x^{3} + 24 x^{2} + 24$$4$$3$$18$$A_4\times C_2$$[2, 2, 2]^{3}$
587Data not computed