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Properties

Label 15.15.1802470272...5361.1
Degree $15$
Signature $[15, 0]$
Discriminant $3^{12}\cdot 13^{8}\cdot 401^{6}$
Root discriminant $104.01$
Ramified primes $3, 13, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T34

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![257049, -3684369, 1685099, 4765293, -2045407, -2429882, 834353, 619723, -143988, -82199, 9555, 5239, -141, -125, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 125*x^13 - 141*x^12 + 5239*x^11 + 9555*x^10 - 82199*x^9 - 143988*x^8 + 619723*x^7 + 834353*x^6 - 2429882*x^5 - 2045407*x^4 + 4765293*x^3 + 1685099*x^2 - 3684369*x + 257049)
 
gp: K = bnfinit(x^15 - 125*x^13 - 141*x^12 + 5239*x^11 + 9555*x^10 - 82199*x^9 - 143988*x^8 + 619723*x^7 + 834353*x^6 - 2429882*x^5 - 2045407*x^4 + 4765293*x^3 + 1685099*x^2 - 3684369*x + 257049, 1)
 

Normalized defining polynomial

\( x^{15} - 125 x^{13} - 141 x^{12} + 5239 x^{11} + 9555 x^{10} - 82199 x^{9} - 143988 x^{8} + 619723 x^{7} + 834353 x^{6} - 2429882 x^{5} - 2045407 x^{4} + 4765293 x^{3} + 1685099 x^{2} - 3684369 x + 257049 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[15, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1802470272605254011884574005361=3^{12}\cdot 13^{8}\cdot 401^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $104.01$
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:  $3, 13, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{13} a^{6} + \frac{5}{13} a^{4} + \frac{2}{13} a^{3}$, $\frac{1}{13} a^{7} + \frac{5}{13} a^{5} + \frac{2}{13} a^{4}$, $\frac{1}{13} a^{8} + \frac{2}{13} a^{5} + \frac{1}{13} a^{4} + \frac{3}{13} a^{3}$, $\frac{1}{169} a^{9} + \frac{5}{169} a^{7} + \frac{2}{169} a^{6} - \frac{2}{13} a^{5} + \frac{4}{13} a^{4} + \frac{4}{13} a^{3}$, $\frac{1}{338} a^{10} + \frac{5}{338} a^{8} + \frac{1}{169} a^{7} - \frac{1}{26} a^{6} - \frac{9}{26} a^{5} + \frac{9}{26} a^{4} + \frac{1}{13} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{1014} a^{11} - \frac{1}{1014} a^{10} - \frac{1}{338} a^{9} - \frac{1}{338} a^{8} - \frac{1}{338} a^{7} + \frac{6}{169} a^{6} + \frac{14}{39} a^{5} + \frac{29}{78} a^{4} - \frac{6}{13} a^{3} - \frac{1}{6} a^{2} - \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{13182} a^{12} + \frac{5}{13182} a^{10} - \frac{4}{2197} a^{9} - \frac{5}{338} a^{8} + \frac{11}{338} a^{7} + \frac{1}{78} a^{6} + \frac{2}{13} a^{5} - \frac{2}{39} a^{4} - \frac{25}{78} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3} a$, $\frac{1}{224094} a^{13} - \frac{1}{74698} a^{12} + \frac{5}{224094} a^{11} + \frac{5}{5746} a^{10} - \frac{15}{74698} a^{9} - \frac{63}{2873} a^{8} + \frac{107}{8619} a^{7} - \frac{61}{5746} a^{6} - \frac{14}{51} a^{5} - \frac{331}{1326} a^{4} + \frac{110}{221} a^{3} - \frac{41}{102} a^{2} + \frac{4}{17} a + \frac{3}{17}$, $\frac{1}{40073844505448721474} a^{14} - \frac{2950947103573}{40073844505448721474} a^{13} + \frac{307323929220391}{20036922252724360737} a^{12} + \frac{5219100947131583}{20036922252724360737} a^{11} + \frac{988695407669650}{742108231582383731} a^{10} - \frac{2540829300413587}{13357948168482907158} a^{9} + \frac{19896050448889081}{3082603423496055498} a^{8} - \frac{32874320730841087}{3082603423496055498} a^{7} + \frac{1857750326904770}{90664806573413397} a^{6} - \frac{6550522637420930}{13173518903829297} a^{5} + \frac{56229996647742896}{118561670134463673} a^{4} - \frac{2136354126487702}{9120128471881821} a^{3} + \frac{65410677449123}{396527324864427} a^{2} - \frac{2198758005437263}{6080085647921214} a - \frac{221597479105927}{1013347607986869}$

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:  $14$
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:  \( 42648761950.9 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T34:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 810
The 24 conjugacy class representatives for [3^4]D(5)
Character table for [3^4]D(5) is not computed

Intermediate fields

5.5.160801.1

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

Sibling fields

Degree 15 siblings: data not computed
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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{3}$ R ${\href{/LocalNumberField/5.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{3}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.6.8.9$x^{6} + 6 x^{5} + 9$$3$$2$$8$$S_3\times C_3$$[2, 2]^{2}$
$13$13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.2$x^{6} - 13 x^{3} + 338$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
401Data not computed