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Properties

Label 15.1.75123889547...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{23}\cdot 5^{15}\cdot 7^{13}\cdot 13^{13}$
Root discriminant $721.75$
Ramified primes $2, 5, 7, 13$
Class number $100$ (GRH)
Class group $[10, 10]$ (GRH)
Galois group $F_5 \times S_3$ (as 15T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-860204310784, 0, 0, 0, 0, 32336668, 0, 0, 0, 0, -12740, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 12740*x^10 + 32336668*x^5 - 860204310784)
 
gp: K = bnfinit(x^15 - 12740*x^10 + 32336668*x^5 - 860204310784, 1)
 

Normalized defining polynomial

\( x^{15} - 12740 x^{10} + 32336668 x^{5} - 860204310784 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-7512388954787337016979704576000000000000000=-\,2^{23}\cdot 5^{15}\cdot 7^{13}\cdot 13^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $721.75$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{7} a^{3}$, $\frac{1}{7} a^{4}$, $\frac{1}{98} a^{5}$, $\frac{1}{98} a^{6}$, $\frac{1}{98} a^{7}$, $\frac{1}{1372} a^{8} - \frac{1}{14} a^{3}$, $\frac{1}{1372} a^{9} - \frac{1}{14} a^{4}$, $\frac{1}{1719836300} a^{10} + \frac{2699}{674975} a^{5} + \frac{1}{13775}$, $\frac{1}{3439672600} a^{11} + \frac{2699}{1349950} a^{6} + \frac{1}{27550} a$, $\frac{1}{626020413200} a^{12} + \frac{1077149}{245690900} a^{7} + \frac{2011151}{5014100} a^{2}$, $\frac{1}{1252040826400} a^{13} + \frac{2699}{491381800} a^{8} - \frac{137749}{10028200} a^{3}$, $\frac{1}{227871430404800} a^{14} + \frac{18626499}{89431487600} a^{9} + \frac{7025251}{1825132400} a^{4}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{10}\times C_{10}$, which has order $100$ (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:  $7$
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:  \( 100428717125459.02 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times F_5$ (as 15T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 120
The 15 conjugacy class representatives for $F_5 \times S_3$
Character table for $F_5 \times S_3$

Intermediate fields

3.1.728.1, 5.1.3428748050000.7

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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R R $15$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$[\ ]_{5}^{4}$
2.10.19.17$x^{10} - 2 x^{4} + 4 x^{2} - 10$$10$$1$$19$$F_{5}\times C_2$$[3]_{5}^{4}$
$5$5.5.5.2$x^{5} + 5 x + 5$$5$$1$$5$$F_5$$[5/4]_{4}$
5.10.10.7$x^{10} + 10 x^{8} + 10 x^{5} - 20 x^{4} - 20 x^{2} + 12$$5$$2$$10$$F_{5}\times C_2$$[5/4]_{4}^{2}$
$7$7.5.4.1$x^{5} - 7$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
7.10.9.1$x^{10} - 7$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
$13$13.5.4.1$x^{5} - 13$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
13.10.9.1$x^{10} - 13$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$