Properties

Label 15.3.6114596477365737.1
Degree 15
Signature $[3, 6]$
Discriminant $3^{4}\cdot 15601\cdot 4838718377$
Ramified primes $3, 15601, 4838718377$
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, -5, 10, -11, 3, 15, -27, 20, 0, -18, 18, -8, -2, 5, -3, 1]);
sage: K = NumberField(x^15 - 3*x^14 + 5*x^13 - 2*x^12 - 8*x^11 + 18*x^10 - 18*x^9 + 20*x^7 - 27*x^6 + 15*x^5 + 3*x^4 - 11*x^3 + 10*x^2 - 5*x + 1,"a")
gp: K = bnfinit(x^15 - 3*x^14 + 5*x^13 - 2*x^12 - 8*x^11 + 18*x^10 - 18*x^9 + 20*x^7 - 27*x^6 + 15*x^5 + 3*x^4 - 11*x^3 + 10*x^2 - 5*x + 1, 1)

Normalized defining polynomial

\(x^{15} \) \(\mathstrut -\mathstrut 3 x^{14} \) \(\mathstrut +\mathstrut 5 x^{13} \) \(\mathstrut -\mathstrut 2 x^{12} \) \(\mathstrut -\mathstrut 8 x^{11} \) \(\mathstrut +\mathstrut 18 x^{10} \) \(\mathstrut -\mathstrut 18 x^{9} \) \(\mathstrut +\mathstrut 20 x^{7} \) \(\mathstrut -\mathstrut 27 x^{6} \) \(\mathstrut +\mathstrut 15 x^{5} \) \(\mathstrut +\mathstrut 3 x^{4} \) \(\mathstrut -\mathstrut 11 x^{3} \) \(\mathstrut +\mathstrut 10 x^{2} \) \(\mathstrut -\mathstrut 5 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(6114596477365737=3^{4}\cdot 15601\cdot 4838718377\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Ramified primes:  $3, 15601, 4838718377$
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:  \( a^{14} - 3 a^{13} + 4 a^{12} - 11 a^{10} + 18 a^{9} - 13 a^{8} - 8 a^{7} + 27 a^{6} - 24 a^{5} + 9 a^{4} + 11 a^{3} - 13 a^{2} + 9 a - 4 \),  \( a \),  \( 4 a^{14} - 9 a^{13} + 13 a^{12} + 2 a^{11} - 30 a^{10} + 47 a^{9} - 33 a^{8} - 26 a^{7} + 56 a^{6} - 56 a^{5} + 10 a^{4} + 20 a^{3} - 22 a^{2} + 16 a - 4 \),  \( 2 a^{14} - 5 a^{13} + 7 a^{12} + a^{11} - 18 a^{10} + 28 a^{9} - 19 a^{8} - 16 a^{7} + 38 a^{6} - 35 a^{5} + 9 a^{4} + 15 a^{3} - 16 a^{2} + 13 a - 5 \),  \( 3 a^{14} - 8 a^{13} + 12 a^{12} - a^{11} - 26 a^{10} + 46 a^{9} - 37 a^{8} - 16 a^{7} + 58 a^{6} - 62 a^{5} + 23 a^{4} + 19 a^{3} - 28 a^{2} + 22 a - 8 \),  \( a^{13} - 2 a^{12} + 2 a^{11} + 3 a^{10} - 10 a^{9} + 11 a^{8} - 2 a^{7} - 15 a^{6} + 19 a^{5} - 11 a^{4} - 5 a^{3} + 10 a^{2} - 8 a + 3 \),  \( 3 a^{14} - 7 a^{13} + 10 a^{12} + 2 a^{11} - 26 a^{10} + 41 a^{9} - 29 a^{8} - 23 a^{7} + 55 a^{6} - 56 a^{5} + 15 a^{4} + 21 a^{3} - 27 a^{2} + 21 a - 7 \),  \( 3 a^{14} - 7 a^{13} + 10 a^{12} + 2 a^{11} - 26 a^{10} + 41 a^{9} - 29 a^{8} - 23 a^{7} + 55 a^{6} - 56 a^{5} + 15 a^{4} + 21 a^{3} - 26 a^{2} + 21 a - 7 \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 53.4601866446 \)
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 $15$ R ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }$ $15$ $15$ $15$ ${\href{/LocalNumberField/17.13.0.1}{13} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ $15$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $15$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ $15$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }$

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$3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
15601Data not computed
4838718377Data not computed