↑

Properties

Label 15.15.6731371672...0000.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{12}\cdot 5^{23}\cdot 13^{10}$
Root discriminant $113.56$
Ramified primes $2, 5, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_5^2 : C_4):C_3$ (as 15T19)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5000, 37500, -93750, 60625, 87500, -106875, -27500, 53125, 3450, -11750, -148, 1245, 0, -60, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 60*x^13 + 1245*x^11 - 148*x^10 - 11750*x^9 + 3450*x^8 + 53125*x^7 - 27500*x^6 - 106875*x^5 + 87500*x^4 + 60625*x^3 - 93750*x^2 + 37500*x - 5000)
 
gp: K = bnfinit(x^15 - 60*x^13 + 1245*x^11 - 148*x^10 - 11750*x^9 + 3450*x^8 + 53125*x^7 - 27500*x^6 - 106875*x^5 + 87500*x^4 + 60625*x^3 - 93750*x^2 + 37500*x - 5000, 1)
 

Normalized defining polynomial

\( x^{15} - 60 x^{13} + 1245 x^{11} - 148 x^{10} - 11750 x^{9} + 3450 x^{8} + 53125 x^{7} - 27500 x^{6} - 106875 x^{5} + 87500 x^{4} + 60625 x^{3} - 93750 x^{2} + 37500 x - 5000 \)

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:  \(6731371672314453125000000000000=2^{12}\cdot 5^{23}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $113.56$
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:  $2, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} + \frac{2}{5} a^{3}$, $\frac{1}{5} a^{9} + \frac{2}{5} a^{4}$, $\frac{1}{25} a^{10} - \frac{1}{5} a^{6} + \frac{2}{25} a^{5} - \frac{1}{5} a^{3}$, $\frac{1}{25} a^{11} - \frac{1}{5} a^{7} + \frac{2}{25} a^{6} - \frac{1}{5} a^{4}$, $\frac{1}{25} a^{12} + \frac{2}{25} a^{7} - \frac{1}{5} a^{5} + \frac{2}{5} a^{3}$, $\frac{1}{1000} a^{13} - \frac{1}{50} a^{12} + \frac{9}{200} a^{9} - \frac{6}{125} a^{8} - \frac{9}{100} a^{7} - \frac{43}{100} a^{6} - \frac{11}{40} a^{5} - \frac{1}{5} a^{4} - \frac{3}{8} a^{3} + \frac{1}{8} a - \frac{1}{4}$, $\frac{1}{8705595228712000} a^{14} - \frac{384344706351}{870559522871200} a^{13} + \frac{240283619493}{43527976143560} a^{12} + \frac{573070896529}{108819940358900} a^{11} + \frac{5453312143257}{1741119045742400} a^{10} - \frac{136826297452649}{4352797614356000} a^{9} + \frac{29934776253783}{870559522871200} a^{8} + \frac{246711457375967}{870559522871200} a^{7} - \frac{476928357255827}{1741119045742400} a^{6} + \frac{216695287694803}{870559522871200} a^{5} - \frac{7312753774011}{69644761829696} a^{4} + \frac{2615334055427}{174111904574240} a^{3} + \frac{28585074159729}{69644761829696} a^{2} + \frac{8344137616439}{17411190457424} a - \frac{5179754595295}{17411190457424}$

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:  \( 62225591181.2 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5^2:C_{12}$ (as 15T19):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 300
The 14 conjugacy class representatives for $(C_5^2 : C_4):C_3$
Character table for $(C_5^2 : C_4):C_3$

Intermediate fields

3.3.169.1

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

Sibling fields

Degree 15 sibling: data not computed
Degree 25 sibling: data not computed
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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ R ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{5}$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.12.12.25$x^{12} - 78 x^{10} - 1621 x^{8} + 460 x^{6} - 1977 x^{4} + 866 x^{2} + 749$$2$$6$$12$$C_{12}$$[2]^{6}$
$5$5.5.9.3$x^{5} + 80$$5$$1$$9$$F_5$$[9/4]_{4}$
5.5.9.2$x^{5} + 55$$5$$1$$9$$F_5$$[9/4]_{4}$
5.5.5.2$x^{5} + 5 x + 5$$5$$1$$5$$F_5$$[5/4]_{4}$
$13$13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.12.8.1$x^{12} - 39 x^{9} - 338 x^{6} + 10985 x^{3} + 228488$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$