Properties

Label 15.3.3276367268581097.1
Degree 15
Signature $[3, 6]$
Discriminant $3276367268581097$
Ramified prime $3276367268581097$
Class number 1
Class group Trivial
Galois Group 15T104

Related objects

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Show commands for: Magma / SageMath / Pari/GP

magma: K<a> := NumberField(PolynomialRing(Rationals())![1, 4, 1, -14, -15, 15, 31, 4, -24, -20, 8, 16, -1, -6, 0, 1]);
sage: K = NumberField(x^15 - 6*x^13 - x^12 + 16*x^11 + 8*x^10 - 20*x^9 - 24*x^8 + 4*x^7 + 31*x^6 + 15*x^5 - 15*x^4 - 14*x^3 + x^2 + 4*x + 1,"a")
gp: K = bnfinit(x^15 - 6*x^13 - x^12 + 16*x^11 + 8*x^10 - 20*x^9 - 24*x^8 + 4*x^7 + 31*x^6 + 15*x^5 - 15*x^4 - 14*x^3 + x^2 + 4*x + 1, 1)

Normalized defining polynomial

\(x^{15} \) \(\mathstrut -\mathstrut 6 x^{13} \) \(\mathstrut -\mathstrut x^{12} \) \(\mathstrut +\mathstrut 16 x^{11} \) \(\mathstrut +\mathstrut 8 x^{10} \) \(\mathstrut -\mathstrut 20 x^{9} \) \(\mathstrut -\mathstrut 24 x^{8} \) \(\mathstrut +\mathstrut 4 x^{7} \) \(\mathstrut +\mathstrut 31 x^{6} \) \(\mathstrut +\mathstrut 15 x^{5} \) \(\mathstrut -\mathstrut 15 x^{4} \) \(\mathstrut -\mathstrut 14 x^{3} \) \(\mathstrut +\mathstrut x^{2} \) \(\mathstrut +\mathstrut 4 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(3276367268581097\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Ramified primes:  $3276367268581097$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{7} a^{13} + \frac{1}{7} a^{12} - \frac{2}{7} a^{11} + \frac{3}{7} a^{9} - \frac{3}{7} a^{8} + \frac{2}{7} a^{6} - \frac{1}{7} a^{5} + \frac{1}{7} a^{4} - \frac{1}{7} a^{3} + \frac{1}{7} a^{2} - \frac{2}{7} a + \frac{2}{7}$, $\frac{1}{7} a^{14} - \frac{3}{7} a^{12} + \frac{2}{7} a^{11} + \frac{3}{7} a^{10} + \frac{1}{7} a^{9} + \frac{3}{7} a^{8} + \frac{2}{7} a^{7} - \frac{3}{7} a^{6} + \frac{2}{7} a^{5} - \frac{2}{7} a^{4} + \frac{2}{7} a^{3} - \frac{3}{7} a^{2} - \frac{3}{7} a - \frac{2}{7}$

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:  \( -1 \)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( a \),  \( \frac{3}{7} a^{14} - \frac{3}{7} a^{13} - \frac{26}{7} a^{12} + \frac{12}{7} a^{11} + \frac{86}{7} a^{10} - \frac{6}{7} a^{9} - \frac{150}{7} a^{8} - \frac{78}{7} a^{7} + \frac{104}{7} a^{6} + \frac{184}{7} a^{5} + \frac{33}{7} a^{4} - \frac{131}{7} a^{3} - \frac{54}{7} a^{2} + \frac{25}{7} a + \frac{16}{7} \),  \( \frac{36}{7} a^{14} - \frac{6}{7} a^{13} - \frac{205}{7} a^{12} + a^{11} + \frac{528}{7} a^{10} + \frac{151}{7} a^{9} - 93 a^{8} - \frac{607}{7} a^{7} + \frac{216}{7} a^{6} + \frac{869}{7} a^{5} + \frac{244}{7} a^{4} - \frac{489}{7} a^{3} - \frac{254}{7} a^{2} + \frac{107}{7} a + 9 \),  \( \frac{13}{7} a^{14} + \frac{4}{7} a^{13} - 11 a^{12} - \frac{31}{7} a^{11} + \frac{200}{7} a^{10} + \frac{137}{7} a^{9} - \frac{225}{7} a^{8} - \frac{317}{7} a^{7} - \frac{10}{7} a^{6} + \frac{344}{7} a^{5} + \frac{216}{7} a^{4} - \frac{139}{7} a^{3} - 20 a^{2} + \frac{9}{7} a + \frac{24}{7} \),  \( \frac{45}{7} a^{14} - \frac{23}{7} a^{13} - \frac{263}{7} a^{12} + \frac{87}{7} a^{11} + \frac{702}{7} a^{10} + \frac{18}{7} a^{9} - \frac{972}{7} a^{8} - \frac{645}{7} a^{7} + \frac{561}{7} a^{6} + \frac{1226}{7} a^{5} + \frac{97}{7} a^{4} - \frac{818}{7} a^{3} - \frac{312}{7} a^{2} + \frac{212}{7} a + \frac{109}{7} \),  \( 6 a^{14} - \frac{15}{7} a^{13} - \frac{239}{7} a^{12} + \frac{51}{7} a^{11} + 88 a^{10} + \frac{74}{7} a^{9} - \frac{788}{7} a^{8} - 86 a^{7} + \frac{355}{7} a^{6} + \frac{995}{7} a^{5} + \frac{146}{7} a^{4} - \frac{594}{7} a^{3} - \frac{225}{7} a^{2} + \frac{142}{7} a + \frac{68}{7} \),  \( \frac{45}{7} a^{14} - \frac{22}{7} a^{13} - \frac{255}{7} a^{12} + \frac{78}{7} a^{11} + \frac{660}{7} a^{10} + 6 a^{9} - \frac{870}{7} a^{8} - \frac{645}{7} a^{7} + \frac{444}{7} a^{6} + 160 a^{5} + 19 a^{4} - 95 a^{3} - \frac{283}{7} a^{2} + 23 a + \frac{90}{7} \),  \( \frac{59}{7} a^{14} - 4 a^{13} - \frac{331}{7} a^{12} + \frac{104}{7} a^{11} + \frac{849}{7} a^{10} + \frac{38}{7} a^{9} - \frac{1104}{7} a^{8} - \frac{792}{7} a^{7} + \frac{565}{7} a^{6} + \frac{1406}{7} a^{5} + \frac{127}{7} a^{4} - \frac{855}{7} a^{3} - \frac{324}{7} a^{2} + \frac{201}{7} a + \frac{99}{7} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 37.7909873347 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

15T104:

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

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $15$ ${\href{/LocalNumberField/3.11.0.1}{11} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ ${\href{/LocalNumberField/5.11.0.1}{11} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.5.0.1}{5} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ $15$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ $15$ $15$ $15$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.11.0.1}{11} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
3276367268581097Data not computed