↑

Properties

Label 15.3.31502030531...7664.1
Degree $15$
Signature $[3, 6]$
Discriminant $2^{16}\cdot 3^{20}\cdot 13^{10}$
Root discriminant $50.11$
Ramified primes $2, 3, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T84

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-11492, -26364, -30420, -26702, -20826, -13923, -6188, -1131, 234, 156, 78, 0, -14, -3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^13 - 14*x^12 + 78*x^10 + 156*x^9 + 234*x^8 - 1131*x^7 - 6188*x^6 - 13923*x^5 - 20826*x^4 - 26702*x^3 - 30420*x^2 - 26364*x - 11492)
 
gp: K = bnfinit(x^15 - 3*x^13 - 14*x^12 + 78*x^10 + 156*x^9 + 234*x^8 - 1131*x^7 - 6188*x^6 - 13923*x^5 - 20826*x^4 - 26702*x^3 - 30420*x^2 - 26364*x - 11492, 1)
 

Normalized defining polynomial

\( x^{15} - 3 x^{13} - 14 x^{12} + 78 x^{10} + 156 x^{9} + 234 x^{8} - 1131 x^{7} - 6188 x^{6} - 13923 x^{5} - 20826 x^{4} - 26702 x^{3} - 30420 x^{2} - 26364 x - 11492 \)

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:  \(31502030531754645746417664=2^{16}\cdot 3^{20}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.11$
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:  $2, 3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{13} a^{11} - \frac{3}{13} a^{9} - \frac{1}{13} a^{8}$, $\frac{1}{13} a^{12} - \frac{3}{13} a^{10} - \frac{1}{13} a^{9}$, $\frac{1}{26} a^{13} - \frac{1}{26} a^{11} + \frac{6}{13} a^{10} - \frac{3}{13} a^{9} - \frac{1}{13} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{8595908433638897230461134} a^{14} - \frac{8237505402993261069235}{661223725664530556189318} a^{13} + \frac{112428203076180829480281}{8595908433638897230461134} a^{12} - \frac{295934926032149745101707}{8595908433638897230461134} a^{11} + \frac{822210492988026798125001}{4297954216819448615230567} a^{10} + \frac{346079427182239518200699}{4297954216819448615230567} a^{9} + \frac{458786078200484366332174}{4297954216819448615230567} a^{8} + \frac{110239573845060654454663}{330611862832265278094659} a^{7} + \frac{171343051660814844353853}{661223725664530556189318} a^{6} - \frac{121613446620315395467395}{661223725664530556189318} a^{5} - \frac{220761517236441377256497}{661223725664530556189318} a^{4} - \frac{5372087370613725990231}{661223725664530556189318} a^{3} - \frac{20595628778420502728070}{330611862832265278094659} a^{2} - \frac{66949062897212654997818}{330611862832265278094659} a + \frac{119896709016394337490986}{330611862832265278094659}$

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:  $8$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 22257278.1186 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T84:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 77760
The 39 conjugacy class representatives for 1/2[S(3)^5]F(5)
Character table for 1/2[S(3)^5]F(5) is not computed

Intermediate fields

5.1.35152.1

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 ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ R ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.10.12.4$x^{10} + 2 x^{5} + 2 x^{3} - 2$$10$$1$$12$$(C_2^4 : C_5):C_4$$[8/5, 8/5, 8/5, 8/5]_{5}^{4}$
3Data not computed
$13$13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.8.7.2$x^{8} - 52$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$