Properties

Label 15.3.2354315214908549.1
Degree 15
Signature $[3, 6]$
Discriminant $27765763\cdot 84792023$
Ramified primes $27765763, 84792023$
Class number 1
Class group Trivial
Galois Group 15T104

Related objects

Learn more about

Show commands for: Magma / SageMath / Pari/GP

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

Normalized defining polynomial

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

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:  \(2354315214908549=27765763\cdot 84792023\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Ramified primes:  $27765763, 84792023$
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 \(a\))

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

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:  \( a \),  \( a^{13} - a^{12} + a^{8} - a^{6} - a^{5} + a^{4} - a^{3} + 1 \),  \( a^{14} - 2 a^{13} + 2 a^{12} - a^{11} - a^{10} + 3 a^{9} - 3 a^{8} + 2 a^{7} - a^{6} - a^{4} + 2 a^{2} - a + 1 \),  \( a^{12} - a^{10} + a^{9} - a^{8} + 2 a^{7} + a^{6} - a^{5} - a^{4} - a^{3} + a^{2} - a + 2 \),  \( a^{13} - a^{12} - a^{9} + 2 a^{8} - a^{6} - a^{5} - a^{3} + a^{2} + a \),  \( a^{14} - 2 a^{13} + a^{11} - a^{10} + 3 a^{9} - 2 a^{8} - 2 a^{7} + 2 a^{5} + a^{4} + a^{2} - 2 a \),  \( a^{14} - a^{13} + a^{11} - 2 a^{10} + 3 a^{9} - a^{8} - a^{7} - a^{5} - a^{3} + 2 a^{2} - 2 a \),  \( a^{14} - a^{13} + a^{12} - 2 a^{10} + 3 a^{9} - 2 a^{8} + a^{7} - a^{5} - a^{3} + 2 a^{2} - 2 a + 2 \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 31.1653663197 \)
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 ${\href{/LocalNumberField/2.9.0.1}{9} }{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }$ ${\href{/LocalNumberField/3.11.0.1}{11} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ $15$ ${\href{/LocalNumberField/7.7.0.1}{7} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.11.0.1}{11} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.9.0.1}{9} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.11.0.1}{11} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.9.0.1}{9} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }$ $15$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$

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
27765763Data not computed
84792023Data not computed