Properties

Label 15.3.2164169497031757.1
Degree 15
Signature $[3, 6]$
Discriminant $3^{3}\cdot 69593\cdot 1151759887$
Ramified primes $3, 69593, 1151759887$
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, -4, 9, -21, 39, -57, 69, -72, 55, -14, -32, 54, -45, 23, -7, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 7*x^14 + 23*x^13 - 45*x^12 + 54*x^11 - 32*x^10 - 14*x^9 + 55*x^8 - 72*x^7 + 69*x^6 - 57*x^5 + 39*x^4 - 21*x^3 + 9*x^2 - 4*x + 1)
gp: K = bnfinit(x^15 - 7*x^14 + 23*x^13 - 45*x^12 + 54*x^11 - 32*x^10 - 14*x^9 + 55*x^8 - 72*x^7 + 69*x^6 - 57*x^5 + 39*x^4 - 21*x^3 + 9*x^2 - 4*x + 1, 1)

Normalized defining polynomial

\(x^{15} \) \(\mathstrut -\mathstrut 7 x^{14} \) \(\mathstrut +\mathstrut 23 x^{13} \) \(\mathstrut -\mathstrut 45 x^{12} \) \(\mathstrut +\mathstrut 54 x^{11} \) \(\mathstrut -\mathstrut 32 x^{10} \) \(\mathstrut -\mathstrut 14 x^{9} \) \(\mathstrut +\mathstrut 55 x^{8} \) \(\mathstrut -\mathstrut 72 x^{7} \) \(\mathstrut +\mathstrut 69 x^{6} \) \(\mathstrut -\mathstrut 57 x^{5} \) \(\mathstrut +\mathstrut 39 x^{4} \) \(\mathstrut -\mathstrut 21 x^{3} \) \(\mathstrut +\mathstrut 9 x^{2} \) \(\mathstrut -\mathstrut 4 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(2164169497031757=3^{3}\cdot 69593\cdot 1151759887\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Ramified primes:  $3, 69593, 1151759887$
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:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( a \),  \( a^{13} - 5 a^{12} + 12 a^{11} - 18 a^{10} + 18 a^{9} - 10 a^{8} - 6 a^{7} + 24 a^{6} - 35 a^{5} + 31 a^{4} - 20 a^{3} + 10 a^{2} - 6 a + 1 \),  \( 3 a^{14} - 18 a^{13} + 50 a^{12} - 80 a^{11} + 70 a^{10} - 8 a^{9} - 68 a^{8} + 107 a^{7} - 103 a^{6} + 80 a^{5} - 56 a^{4} + 30 a^{3} - 13 a^{2} + 4 a - 2 \),  \( a^{14} - 7 a^{13} + 22 a^{12} - 40 a^{11} + 43 a^{10} - 20 a^{9} - 16 a^{8} + 43 a^{7} - 55 a^{6} + 58 a^{5} - 51 a^{4} + 34 a^{3} - 17 a^{2} + 8 a - 3 \),  \( a^{14} - 5 a^{13} + 11 a^{12} - 12 a^{11} + 3 a^{10} + 8 a^{9} - 10 a^{8} + 5 a^{7} - 6 a^{6} + 12 a^{5} - 14 a^{4} + 8 a^{3} - 5 a^{2} + 3 a - 2 \),  \( a^{13} - 7 a^{12} + 21 a^{11} - 34 a^{10} + 26 a^{9} + 8 a^{8} - 42 a^{7} + 48 a^{6} - 31 a^{5} + 17 a^{4} - 11 a^{3} + 5 a^{2} + a \),  \( a^{14} - 5 a^{13} + 11 a^{12} - 11 a^{11} - 3 a^{10} + 24 a^{9} - 33 a^{8} + 20 a^{7} + a^{6} - 15 a^{5} + 17 a^{4} - 15 a^{3} + 9 a^{2} - 5 a + 1 \),  \( a^{14} - 5 a^{13} + 12 a^{12} - 18 a^{11} + 18 a^{10} - 11 a^{9} - 2 a^{8} + 18 a^{7} - 32 a^{6} + 35 a^{5} - 28 a^{4} + 17 a^{3} - 10 a^{2} + 3 a - 1 \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 29.6657425505 \)
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 ${\href{/LocalNumberField/2.9.0.1}{9} }{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }$ R ${\href{/LocalNumberField/5.9.0.1}{9} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }$ ${\href{/LocalNumberField/7.13.0.1}{13} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.11.0.1}{11} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ ${\href{/LocalNumberField/13.13.0.1}{13} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.7.0.1}{7} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }$

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$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.9.0.1$x^{9} - x^{3} + x^{2} + 1$$1$$9$$0$$C_9$$[\ ]^{9}$
69593Data not computed
1151759887Data not computed