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Properties

Label 15.13.1726328604...0000.1
Degree $15$
Signature $[13, 1]$
Discriminant $-\,2^{12}\cdot 3^{19}\cdot 5^{15}\cdot 7^{6}\cdot 101$
Root discriminant $103.71$
Ramified primes $2, 3, 5, 7, 101$
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![-9587, 62925, -74970, -65975, 89985, 41355, -30560, -15165, -90, 1230, 1320, 450, -75, -45, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 45*x^13 - 75*x^12 + 450*x^11 + 1320*x^10 + 1230*x^9 - 90*x^8 - 15165*x^7 - 30560*x^6 + 41355*x^5 + 89985*x^4 - 65975*x^3 - 74970*x^2 + 62925*x - 9587)
 
gp: K = bnfinit(x^15 - 45*x^13 - 75*x^12 + 450*x^11 + 1320*x^10 + 1230*x^9 - 90*x^8 - 15165*x^7 - 30560*x^6 + 41355*x^5 + 89985*x^4 - 65975*x^3 - 74970*x^2 + 62925*x - 9587, 1)
 

Normalized defining polynomial

\( x^{15} - 45 x^{13} - 75 x^{12} + 450 x^{11} + 1320 x^{10} + 1230 x^{9} - 90 x^{8} - 15165 x^{7} - 30560 x^{6} + 41355 x^{5} + 89985 x^{4} - 65975 x^{3} - 74970 x^{2} + 62925 x - 9587 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[13, 1]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1726328604054922875000000000000=-\,2^{12}\cdot 3^{19}\cdot 5^{15}\cdot 7^{6}\cdot 101\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $103.71$
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:  $2, 3, 5, 7, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{31} a^{13} + \frac{6}{31} a^{12} + \frac{13}{31} a^{11} + \frac{11}{31} a^{10} - \frac{4}{31} a^{9} - \frac{12}{31} a^{8} - \frac{15}{31} a^{7} - \frac{10}{31} a^{6} + \frac{7}{31} a^{5} + \frac{14}{31} a^{4} - \frac{9}{31} a^{3} - \frac{2}{31} a^{2} + \frac{6}{31}$, $\frac{1}{21648606476162197721632323} a^{14} - \frac{87320315269412037864181}{21648606476162197721632323} a^{13} - \frac{5634087087741449659685954}{21648606476162197721632323} a^{12} - \frac{10290510553067965887589330}{21648606476162197721632323} a^{11} - \frac{4003607782267422070790819}{21648606476162197721632323} a^{10} - \frac{148138187365651336137313}{698342144392328958762333} a^{9} - \frac{4135318327134459056579921}{21648606476162197721632323} a^{8} + \frac{695760843820255937897267}{21648606476162197721632323} a^{7} + \frac{1996580352345641195395915}{21648606476162197721632323} a^{6} - \frac{1991926928444259799932166}{7216202158720732573877441} a^{5} - \frac{2311790608367134019141026}{7216202158720732573877441} a^{4} + \frac{1223742186186038392788350}{7216202158720732573877441} a^{3} + \frac{2043603438755290836376576}{21648606476162197721632323} a^{2} + \frac{2617948117871219898330509}{21648606476162197721632323} a + \frac{4773745420162733986786000}{21648606476162197721632323}$

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:  $13$
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:  \( 30584458462.7 \) (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

5.5.2450000.1

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 R R R $15$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ $15$

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
2Data not computed
$3$3.3.3.1$x^{3} + 6 x + 3$$3$$1$$3$$S_3$$[3/2]_{2}$
3.12.16.46$x^{12} + 30 x^{11} - 243 x^{10} + 345 x^{9} - 126 x^{8} + 135 x^{7} - 261 x^{6} - 54 x^{5} + 81 x^{4} - 243 x^{3} - 81 x^{2} + 243 x + 324$$3$$4$$16$12T72$[2, 2, 2]^{4}$
$5$5.5.5.1$x^{5} + 20 x + 5$$5$$1$$5$$F_5$$[5/4]_{4}$
5.10.10.10$x^{10} + 10 x^{8} + 5 x^{6} + 10 x^{5} - 20 x^{4} - 20 x^{2} + 2$$5$$2$$10$$F_{5}\times C_2$$[5/4]_{4}^{2}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.12.6.2$x^{12} + 7203 x^{4} - 16807 x^{2} + 588245$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
101Data not computed