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Properties

Label 15.5.12949340997...0384.1
Degree $15$
Signature $[5, 5]$
Discriminant $-\,2^{16}\cdot 37\cdot 53^{9}\cdot 161839$
Root discriminant $64.19$
Ramified primes $2, 37, 53, 161839$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T87

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-16, 0, 80, -80, -124, 248, -64, -180, 175, -40, -31, 25, -15, 10, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 + 10*x^13 - 15*x^12 + 25*x^11 - 31*x^10 - 40*x^9 + 175*x^8 - 180*x^7 - 64*x^6 + 248*x^5 - 124*x^4 - 80*x^3 + 80*x^2 - 16)
 
gp: K = bnfinit(x^15 - 5*x^14 + 10*x^13 - 15*x^12 + 25*x^11 - 31*x^10 - 40*x^9 + 175*x^8 - 180*x^7 - 64*x^6 + 248*x^5 - 124*x^4 - 80*x^3 + 80*x^2 - 16, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} + 10 x^{13} - 15 x^{12} + 25 x^{11} - 31 x^{10} - 40 x^{9} + 175 x^{8} - 180 x^{7} - 64 x^{6} + 248 x^{5} - 124 x^{4} - 80 x^{3} + 80 x^{2} - 16 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[5, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1294934099725229745485840384=-\,2^{16}\cdot 37\cdot 53^{9}\cdot 161839\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.19$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 37, 53, 161839$
magma: PrimeDivisors(Discriminant(Integers(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 \(a\))

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

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
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:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 90894315.266 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T87:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 155520
The 63 conjugacy class representatives for [S(3)^5]F(5)=S(3)wrF(5) are not computed
Character table for [S(3)^5]F(5)=S(3)wrF(5) is not computed

Intermediate fields

5.5.2382032.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 30 siblings: 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 R ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ R ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ R ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$

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
$2$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.8.12.19$x^{8} + 12 x^{4} + 80$$4$$2$$12$$(C_8:C_2):C_2$$[2, 2, 2]^{4}$
37Data not computed
$53$$\Q_{53}$$x + 2$$1$$1$$0$Trivial$[\ ]$
53.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
161839Data not computed