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Properties

Label 15.3.91113700399...2208.1
Degree $15$
Signature $[3, 6]$
Discriminant $2^{10}\cdot 13^{3}\cdot 347^{3}\cdot 751\cdot 12907$
Root discriminant $24.96$
Ramified primes $2, 13, 347, 751, 12907$
Class number $1$
Class group Trivial
Galois group 15T93

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-32, 0, 64, -112, 8, 32, -68, 36, -2, -2, -7, 10, 0, -1, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^13 + 10*x^11 - 7*x^10 - 2*x^9 - 2*x^8 + 36*x^7 - 68*x^6 + 32*x^5 + 8*x^4 - 112*x^3 + 64*x^2 - 32)
 
gp: K = bnfinit(x^15 - x^13 + 10*x^11 - 7*x^10 - 2*x^9 - 2*x^8 + 36*x^7 - 68*x^6 + 32*x^5 + 8*x^4 - 112*x^3 + 64*x^2 - 32, 1)
 

Normalized defining polynomial

\( x^{15} - x^{13} + 10 x^{11} - 7 x^{10} - 2 x^{9} - 2 x^{8} + 36 x^{7} - 68 x^{6} + 32 x^{5} + 8 x^{4} - 112 x^{3} + 64 x^{2} - 32 \)

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:  \(911137003994932702208=2^{10}\cdot 13^{3}\cdot 347^{3}\cdot 751\cdot 12907\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.96$
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:  $2, 13, 347, 751, 12907$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{9} - \frac{1}{4} a^{7} + \frac{1}{8} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{10} - \frac{1}{4} a^{8} + \frac{1}{8} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{11} - \frac{1}{8} a^{9} + \frac{1}{16} a^{8} + \frac{1}{8} a^{7} + \frac{3}{8} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{650291632} a^{14} + \frac{616597}{81286454} a^{13} - \frac{8074185}{650291632} a^{12} + \frac{10869091}{325145816} a^{11} + \frac{34470209}{325145816} a^{10} - \frac{12306873}{650291632} a^{9} - \frac{23950769}{325145816} a^{8} + \frac{67186659}{325145816} a^{7} - \frac{123674563}{325145816} a^{6} - \frac{23967669}{81286454} a^{5} + \frac{10943807}{40643227} a^{4} + \frac{18657771}{40643227} a^{3} + \frac{8978912}{40643227} a^{2} + \frac{4996466}{40643227} a - \frac{11715454}{40643227}$

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

Class group and class number

Trivial group, which has 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:  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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 33474.0949441 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T93:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 933120
The 108 conjugacy class representatives for [S(3)^5]S(5)=S(3)wrS(5) are not computed
Character table for [S(3)^5]S(5)=S(3)wrS(5) is not computed

Intermediate fields

5.3.4511.1

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
Degree 45 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 R ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }$ ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }$ $15$ ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ R ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.9.0.1}{9} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.14$x^{10} + 5 x^{8} - 50 x^{6} - 58 x^{4} + 49 x^{2} + 21$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
347Data not computed
751Data not computed
12907Data not computed