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Properties

Label 15.15.1008471948...0000.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{12}\cdot 5^{15}\cdot 7^{10}\cdot 13^{4}$
Root discriminant $63.13$
Ramified primes $2, 5, 7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T38

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1352, 5980, -26650, 14105, 32500, -29575, -12460, 16825, 1870, -4450, -92, 605, 0, -40, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 40*x^13 + 605*x^11 - 92*x^10 - 4450*x^9 + 1870*x^8 + 16825*x^7 - 12460*x^6 - 29575*x^5 + 32500*x^4 + 14105*x^3 - 26650*x^2 + 5980*x + 1352)
 
gp: K = bnfinit(x^15 - 40*x^13 + 605*x^11 - 92*x^10 - 4450*x^9 + 1870*x^8 + 16825*x^7 - 12460*x^6 - 29575*x^5 + 32500*x^4 + 14105*x^3 - 26650*x^2 + 5980*x + 1352, 1)
 

Normalized defining polynomial

\( x^{15} - 40 x^{13} + 605 x^{11} - 92 x^{10} - 4450 x^{9} + 1870 x^{8} + 16825 x^{7} - 12460 x^{6} - 29575 x^{5} + 32500 x^{4} + 14105 x^{3} - 26650 x^{2} + 5980 x + 1352 \)

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:  \(1008471948336125000000000000=2^{12}\cdot 5^{15}\cdot 7^{10}\cdot 13^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $63.13$
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:  $2, 5, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{16} a^{13} - \frac{1}{4} a^{12} + \frac{1}{4} a^{11} - \frac{3}{16} a^{9} - \frac{3}{8} a^{7} + \frac{3}{8} a^{6} - \frac{7}{16} a^{5} - \frac{1}{2} a^{4} + \frac{5}{16} a^{3} + \frac{5}{16} a + \frac{1}{8}$, $\frac{1}{49301255381248} a^{14} + \frac{960443235}{1896202130048} a^{13} - \frac{690268101273}{12325313845312} a^{12} + \frac{236094189461}{474050532512} a^{11} - \frac{1641219559219}{49301255381248} a^{10} - \frac{5640413002387}{24650627690624} a^{9} - \frac{9366176544571}{24650627690624} a^{8} - \frac{8403228899859}{24650627690624} a^{7} - \frac{7991008580955}{49301255381248} a^{6} + \frac{8429978445613}{24650627690624} a^{5} - \frac{1567782481991}{3792404260096} a^{4} - \frac{759934123663}{1896202130048} a^{3} - \frac{623342569831}{3792404260096} a^{2} - \frac{24123383401}{948101065024} a - \frac{39239855243}{948101065024}$

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:  \( 557433481.465 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T38:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1500
The 25 conjugacy class representatives for [5^3:4]3
Character table for [5^3:4]3 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\)

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

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 R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ R ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ $15$

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$2.3.0.1$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}$
5Data not computed
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.5.4.1$x^{5} - 13$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$