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Properties

Label 15.5.36495998433...7104.1
Degree $15$
Signature $[5, 5]$
Discriminant $-\,2^{10}\cdot 11^{12}\cdot 11356201$
Root discriminant $31.93$
Ramified primes $2, 11, 11356201$
Class number $1$
Class group Trivial
Galois group 15T81

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![32, 80, 80, -56, -182, -143, -56, -18, -6, 39, 40, 10, -12, -6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 6*x^13 - 12*x^12 + 10*x^11 + 40*x^10 + 39*x^9 - 6*x^8 - 18*x^7 - 56*x^6 - 143*x^5 - 182*x^4 - 56*x^3 + 80*x^2 + 80*x + 32)
 
gp: K = bnfinit(x^15 - 6*x^13 - 12*x^12 + 10*x^11 + 40*x^10 + 39*x^9 - 6*x^8 - 18*x^7 - 56*x^6 - 143*x^5 - 182*x^4 - 56*x^3 + 80*x^2 + 80*x + 32, 1)
 

Normalized defining polynomial

\( x^{15} - 6 x^{13} - 12 x^{12} + 10 x^{11} + 40 x^{10} + 39 x^{9} - 6 x^{8} - 18 x^{7} - 56 x^{6} - 143 x^{5} - 182 x^{4} - 56 x^{3} + 80 x^{2} + 80 x + 32 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[5, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-36495998433430934447104=-\,2^{10}\cdot 11^{12}\cdot 11356201\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.93$
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:  $2, 11, 11356201$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{8} a^{11} + \frac{1}{4} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} - \frac{1}{8} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{8} a + \frac{1}{4}$, $\frac{1}{64} a^{12} + \frac{1}{32} a^{11} + \frac{9}{32} a^{10} + \frac{3}{8} a^{9} + \frac{9}{32} a^{8} + \frac{7}{16} a^{7} + \frac{7}{64} a^{6} - \frac{3}{8} a^{5} - \frac{11}{32} a^{4} - \frac{7}{16} a^{3} + \frac{1}{64} a^{2} + \frac{3}{16} a + \frac{5}{16}$, $\frac{1}{512} a^{13} - \frac{1}{128} a^{12} + \frac{3}{256} a^{11} - \frac{5}{128} a^{10} + \frac{33}{256} a^{9} - \frac{9}{32} a^{8} - \frac{161}{512} a^{7} + \frac{95}{256} a^{6} - \frac{67}{256} a^{5} + \frac{29}{64} a^{4} - \frac{23}{512} a^{3} + \frac{3}{256} a^{2} + \frac{3}{128} a + \frac{1}{64}$, $\frac{1}{4096} a^{14} - \frac{1}{2048} a^{13} - \frac{1}{2048} a^{12} - \frac{1}{512} a^{11} + \frac{13}{2048} a^{10} - \frac{3}{1024} a^{9} + \frac{63}{4096} a^{8} - \frac{33}{1024} a^{7} + \frac{123}{2048} a^{6} - \frac{137}{1024} a^{5} + \frac{953}{4096} a^{4} + \frac{251}{512} a^{3} + \frac{3}{512} a^{2} + \frac{1}{128} a + \frac{1}{256}$

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:  $9$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 302797.187809 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T81:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 38880
The 63 conjugacy class representatives for [S(3)^5]5=S(3)wr5 are not computed
Character table for [S(3)^5]5=S(3)wr5 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\)

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 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 ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }$ $15$ $15$ R $15$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ $15$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }$ $15$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }$ $15$

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$$[\ ]^{5}$
2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11356201Data not computed