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Properties

Label 15.1.98862321753...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{23}\cdot 3^{13}\cdot 5^{17}\cdot 7^{13}$
Root discriminant $251.00$
Ramified primes $2, 3, 5, 7$
Class number $25$ (GRH)
Class group $[5, 5]$ (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![398297088, 0, 0, 0, 0, 12993792, 0, 0, 0, 0, -5124, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5124*x^10 + 12993792*x^5 + 398297088)
 
gp: K = bnfinit(x^15 - 5124*x^10 + 12993792*x^5 + 398297088, 1)
 

Normalized defining polynomial

\( x^{15} - 5124 x^{10} + 12993792 x^{5} + 398297088 \)

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:  \(-988623217530364550400000000000000000=-\,2^{23}\cdot 3^{13}\cdot 5^{17}\cdot 7^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $251.00$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{20} a^{5} - \frac{2}{5}$, $\frac{1}{20} a^{6} - \frac{2}{5} a$, $\frac{1}{40} a^{7} + \frac{3}{10} a^{2}$, $\frac{1}{80} a^{8} + \frac{3}{20} a^{3}$, $\frac{1}{800} a^{9} - \frac{1}{400} a^{8} + \frac{1}{200} a^{7} - \frac{1}{100} a^{6} + \frac{1}{50} a^{5} + \frac{23}{200} a^{4} - \frac{23}{100} a^{3} + \frac{23}{50} a^{2} + \frac{2}{25} a - \frac{4}{25}$, $\frac{1}{52382400} a^{10} - \frac{2049}{623600} a^{5} - \frac{2194}{38975}$, $\frac{1}{104764800} a^{11} + \frac{29131}{1247200} a^{6} - \frac{8892}{38975} a$, $\frac{1}{22000608000} a^{12} + \frac{1}{261912000} a^{11} + \frac{1}{261912000} a^{10} + \frac{1089251}{261912000} a^{7} - \frac{2049}{3118000} a^{6} - \frac{2049}{3118000} a^{5} + \frac{1461014}{4092375} a^{2} - \frac{41169}{194875} a - \frac{41169}{194875}$, $\frac{1}{44001216000} a^{13} + \frac{1}{261912000} a^{11} - \frac{1}{130956000} a^{10} + \frac{1089251}{523824000} a^{8} - \frac{2049}{3118000} a^{6} + \frac{2049}{1559000} a^{5} + \frac{730507}{4092375} a^{3} - \frac{41169}{194875} a + \frac{82338}{194875}$, $\frac{1}{1848051072000} a^{14} - \frac{1}{523824000} a^{11} - \frac{1}{130956000} a^{10} - \frac{4148989}{22000608000} a^{9} - \frac{1}{400} a^{8} + \frac{1}{200} a^{7} + \frac{95589}{6236000} a^{6} + \frac{33229}{1559000} a^{5} + \frac{6208169}{343759500} a^{4} - \frac{23}{100} a^{3} + \frac{23}{50} a^{2} + \frac{94637}{194875} a + \frac{51158}{194875}$

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

Class group and class number

$C_{5}\times C_{5}$, which has order $25$ (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:  \( 248951624693.55133 \) (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

3.1.4200.1, 5.1.9724050000.16

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 ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$$[\ ]_{5}^{4}$
2.10.19.49$x^{10} - 6$$10$$1$$19$$F_{5}\times C_2$$[3]_{5}^{4}$
$3$3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.10.9.1$x^{10} - 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
$5$5.15.17.3$x^{15} - 5 x^{13} + 10 x^{12} + 10 x^{11} + 10 x^{9} - 5 x^{8} - 10 x^{6} + 10 x^{5} + 5 x^{3} + 10$$15$$1$$17$$F_5 \times S_3$$[5/4]_{12}^{2}$
$7$7.5.4.1$x^{5} - 7$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
7.10.9.1$x^{10} - 7$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$