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Properties

Label 15.3.27799923034...0000.1
Degree $15$
Signature $[3, 6]$
Discriminant $2^{10}\cdot 3^{12}\cdot 5^{6}\cdot 83^{6}$
Root discriminant $42.62$
Ramified primes $2, 3, 5, 83$
Class number $1$
Class group Trivial
Galois group 15T80

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3296, -2928, 5328, 760, -4038, -27, 1240, 333, -324, -262, 72, 90, -6, -15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 15*x^13 - 6*x^12 + 90*x^11 + 72*x^10 - 262*x^9 - 324*x^8 + 333*x^7 + 1240*x^6 - 27*x^5 - 4038*x^4 + 760*x^3 + 5328*x^2 - 2928*x - 3296)
 
gp: K = bnfinit(x^15 - 15*x^13 - 6*x^12 + 90*x^11 + 72*x^10 - 262*x^9 - 324*x^8 + 333*x^7 + 1240*x^6 - 27*x^5 - 4038*x^4 + 760*x^3 + 5328*x^2 - 2928*x - 3296, 1)
 

Normalized defining polynomial

\( x^{15} - 15 x^{13} - 6 x^{12} + 90 x^{11} + 72 x^{10} - 262 x^{9} - 324 x^{8} + 333 x^{7} + 1240 x^{6} - 27 x^{5} - 4038 x^{4} + 760 x^{3} + 5328 x^{2} - 2928 x - 3296 \)

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:  \(2779992303417515664000000=2^{10}\cdot 3^{12}\cdot 5^{6}\cdot 83^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.62$
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:  $2, 3, 5, 83$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{3} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{1}{8} a^{3} + \frac{1}{8} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{16} a^{6} - \frac{1}{8} a^{4} - \frac{3}{16} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{16} a^{7} - \frac{1}{8} a^{4} - \frac{1}{16} a^{3} + \frac{1}{8} a^{2}$, $\frac{1}{32} a^{8} - \frac{1}{32} a^{7} - \frac{1}{16} a^{5} - \frac{3}{32} a^{4} - \frac{1}{32} a^{3} - \frac{3}{16} a^{2} + \frac{1}{8} a + \frac{1}{4}$, $\frac{1}{64} a^{9} - \frac{1}{64} a^{7} - \frac{1}{32} a^{6} + \frac{3}{64} a^{5} + \frac{1}{16} a^{4} + \frac{1}{64} a^{3} - \frac{5}{32} a^{2} - \frac{1}{16} a + \frac{1}{8}$, $\frac{1}{64} a^{10} - \frac{1}{64} a^{8} - \frac{1}{32} a^{7} - \frac{1}{64} a^{6} - \frac{1}{16} a^{5} + \frac{1}{64} a^{4} - \frac{1}{32} a^{3} - \frac{1}{4} a^{2} - \frac{3}{8} a - \frac{1}{4}$, $\frac{1}{128} a^{11} - \frac{1}{128} a^{10} - \frac{1}{128} a^{9} - \frac{1}{128} a^{8} - \frac{3}{128} a^{7} + \frac{1}{128} a^{6} - \frac{3}{128} a^{5} + \frac{5}{128} a^{4} + \frac{7}{64} a^{3} - \frac{1}{32} a^{2} - \frac{1}{16} a$, $\frac{1}{1280} a^{12} + \frac{1}{160} a^{10} - \frac{3}{640} a^{8} - \frac{1}{32} a^{6} + \frac{13}{1280} a^{4} - \frac{31}{160} a^{2} + \frac{1}{80}$, $\frac{1}{3840} a^{13} - \frac{1}{3840} a^{12} + \frac{1}{480} a^{11} + \frac{1}{320} a^{10} + \frac{7}{1920} a^{9} - \frac{7}{1920} a^{8} + \frac{1}{64} a^{7} - \frac{1}{192} a^{6} - \frac{7}{3840} a^{5} + \frac{29}{1280} a^{4} + \frac{53}{960} a^{3} - \frac{17}{240} a^{2} + \frac{43}{120} a - \frac{91}{240}$, $\frac{1}{7680} a^{14} - \frac{1}{7680} a^{13} + \frac{1}{3840} a^{12} + \frac{1}{640} a^{11} - \frac{17}{3840} a^{10} - \frac{7}{3840} a^{9} + \frac{1}{80} a^{8} + \frac{11}{384} a^{7} + \frac{233}{7680} a^{6} + \frac{29}{2560} a^{5} - \frac{173}{3840} a^{4} + \frac{7}{120} a^{3} - \frac{31}{480} a^{2} + \frac{209}{480} a - \frac{21}{80}$

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:  \( 2496127.10874 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T80:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 38880
The 48 conjugacy class representatives for [1/2.S(3)^5]D(5)
Character table for [1/2.S(3)^5]D(5) is not computed

Intermediate fields

5.1.172225.1

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 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 R R ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ $15$ $15$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $15$ $15$ $15$ $15$ $15$

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$$[\ ]^{5}$
2.10.10.3$x^{10} - 9 x^{8} + 22 x^{6} - 46 x^{4} + 9 x^{2} - 9$$2$$5$$10$$C_2^4 : C_5$$[2, 2, 2, 2]^{5}$
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.3.4.4$x^{3} + 3 x^{2} + 3$$3$$1$$4$$S_3$$[2]^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.6.6.5$x^{6} + 6 x^{3} + 9 x^{2} + 9$$3$$2$$6$$S_3^2$$[3/2, 3/2]_{2}^{2}$
$5$5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$83$83.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
83.6.3.1$x^{6} - 166 x^{4} + 6889 x^{2} - 5146083$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
83.6.3.1$x^{6} - 166 x^{4} + 6889 x^{2} - 5146083$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$