Properties

Label 15.1.5976545641547631.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,3\cdot 195479\cdot 10191283363$
Root discriminant $11.27$
Ramified primes $3, 195479, 10191283363$
Class number $1$
Class group Trivial
Galois Group 15T104

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 6, -15, 21, -20, 18, -20, 22, -22, 23, -20, 13, -10, 5, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 + 5*x^13 - 10*x^12 + 13*x^11 - 20*x^10 + 23*x^9 - 22*x^8 + 22*x^7 - 20*x^6 + 18*x^5 - 20*x^4 + 21*x^3 - 15*x^2 + 6*x - 1)
gp: K = bnfinit(x^15 - 2*x^14 + 5*x^13 - 10*x^12 + 13*x^11 - 20*x^10 + 23*x^9 - 22*x^8 + 22*x^7 - 20*x^6 + 18*x^5 - 20*x^4 + 21*x^3 - 15*x^2 + 6*x - 1, 1)

Normalized defining polynomial

\(x^{15} \) \(\mathstrut -\mathstrut 2 x^{14} \) \(\mathstrut +\mathstrut 5 x^{13} \) \(\mathstrut -\mathstrut 10 x^{12} \) \(\mathstrut +\mathstrut 13 x^{11} \) \(\mathstrut -\mathstrut 20 x^{10} \) \(\mathstrut +\mathstrut 23 x^{9} \) \(\mathstrut -\mathstrut 22 x^{8} \) \(\mathstrut +\mathstrut 22 x^{7} \) \(\mathstrut -\mathstrut 20 x^{6} \) \(\mathstrut +\mathstrut 18 x^{5} \) \(\mathstrut -\mathstrut 20 x^{4} \) \(\mathstrut +\mathstrut 21 x^{3} \) \(\mathstrut -\mathstrut 15 x^{2} \) \(\mathstrut +\mathstrut 6 x \) \(\mathstrut -\mathstrut 1 \)

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:  \(-5976545641547631=-\,3\cdot 195479\cdot 10191283363\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.27$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 195479, 10191283363$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{61} a^{14} + \frac{28}{61} a^{13} - \frac{9}{61} a^{12} + \frac{25}{61} a^{11} - \frac{30}{61} a^{10} - \frac{5}{61} a^{9} - \frac{5}{61} a^{8} + \frac{11}{61} a^{7} - \frac{14}{61} a^{6} - \frac{13}{61} a^{5} - \frac{6}{61} a^{4} - \frac{17}{61} a^{3} - \frac{1}{61} a^{2} + \frac{16}{61} a - \frac{2}{61}$

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:  \( a \),  \( \frac{524}{61} a^{14} - \frac{578}{61} a^{13} + \frac{2055}{61} a^{12} - \frac{3370}{61} a^{11} + \frac{3617}{61} a^{10} - \frac{7012}{61} a^{9} + \frac{5554}{61} a^{8} - \frac{6009}{61} a^{7} + \frac{5901}{61} a^{6} - \frac{4799}{61} a^{5} + \frac{4786}{61} a^{4} - \frac{5980}{61} a^{3} + \frac{5393}{61} a^{2} - \frac{2596}{61} a + \frac{477}{61} \),  \( \frac{485}{61} a^{14} - \frac{755}{61} a^{13} + \frac{2101}{61} a^{12} - \frac{3918}{61} a^{11} + \frac{4604}{61} a^{10} - \frac{7671}{61} a^{9} + \frac{7762}{61} a^{8} - \frac{7292}{61} a^{7} + \frac{7362}{61} a^{6} - \frac{6488}{61} a^{5} + \frac{5813}{61} a^{4} - \frac{7086}{61} a^{3} + \frac{7079}{61} a^{2} - \frac{4135}{61} a + \frac{1043}{61} \),  \( \frac{128}{61} a^{14} - \frac{76}{61} a^{13} + \frac{434}{61} a^{12} - \frac{582}{61} a^{11} + \frac{491}{61} a^{10} - \frac{1311}{61} a^{9} + \frac{580}{61} a^{8} - \frac{849}{61} a^{7} + \frac{831}{61} a^{6} - \frac{566}{61} a^{5} + \frac{574}{61} a^{4} - \frac{1017}{61} a^{3} + \frac{604}{61} a^{2} - \frac{26}{61} a - \frac{73}{61} \),  \( \frac{1305}{61} a^{14} - \frac{1890}{61} a^{13} + \frac{5457}{61} a^{12} - \frac{10014}{61} a^{11} + \frac{11358}{61} a^{10} - \frac{19701}{61} a^{9} + \frac{19034}{61} a^{8} - \frac{17975}{61} a^{7} + \frac{18635}{61} a^{6} - \frac{15684}{61} a^{5} + \frac{14679}{61} a^{4} - \frac{17854}{61} a^{3} + \frac{17422}{61} a^{2} - \frac{9742}{61} a + \frac{2270}{61} \),  \( \frac{2073}{61} a^{14} - \frac{2895}{61} a^{13} + \frac{8610}{61} a^{12} - \frac{15519}{61} a^{11} + \frac{17537}{61} a^{10} - \frac{30800}{61} a^{9} + \frac{28980}{61} a^{8} - \frac{27949}{61} a^{7} + \frac{28562}{61} a^{6} - \frac{24021}{61} a^{5} + \frac{22637}{61} a^{4} - \frac{27616}{61} a^{3} + \frac{26719}{61} a^{2} - \frac{14778}{61} a + \frac{3357}{61} \),  \( \frac{133}{61} a^{14} - \frac{58}{61} a^{13} + \frac{450}{61} a^{12} - \frac{518}{61} a^{11} + \frac{463}{61} a^{10} - \frac{1275}{61} a^{9} + \frac{372}{61} a^{8} - \frac{916}{61} a^{7} + \frac{578}{61} a^{6} - \frac{570}{61} a^{5} + \frac{544}{61} a^{4} - \frac{858}{61} a^{3} + \frac{538}{61} a^{2} - \frac{7}{61} a - \frac{144}{61} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 50.8430332143 \)
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
The 176 conjugacy class representatives for S15 are not computed
Character table for S15 is not computed

Intermediate fields

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

Sibling fields

Degree 30 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 $15$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ $15$ $15$ ${\href{/LocalNumberField/13.7.0.1}{7} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ $15$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.14.0.1}{14} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ $15$ $15$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$

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
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.9.0.1$x^{9} - x^{3} + x^{2} + 1$$1$$9$$0$$C_9$$[\ ]^{9}$
195479Data not computed
10191283363Data not computed