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Properties

Label 15.15.1435834106...3125.1
Degree $15$
Signature $[15, 0]$
Discriminant $5^{5}\cdot 61^{3}\cdot 397^{3}\cdot 5687830021^{2}$
Root discriminant $257.32$
Ramified primes $5, 61, 397, 5687830021$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 15T83

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-291503125, -858165625, -999059375, -536596250, -77255625, 53553625, 24552875, 698150, -1521725, -228980, 37099, 9363, -322, -157, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 157*x^13 - 322*x^12 + 9363*x^11 + 37099*x^10 - 228980*x^9 - 1521725*x^8 + 698150*x^7 + 24552875*x^6 + 53553625*x^5 - 77255625*x^4 - 536596250*x^3 - 999059375*x^2 - 858165625*x - 291503125)
 
gp: K = bnfinit(x^15 - 157*x^13 - 322*x^12 + 9363*x^11 + 37099*x^10 - 228980*x^9 - 1521725*x^8 + 698150*x^7 + 24552875*x^6 + 53553625*x^5 - 77255625*x^4 - 536596250*x^3 - 999059375*x^2 - 858165625*x - 291503125, 1)
 

Normalized defining polynomial

\( x^{15} - 157 x^{13} - 322 x^{12} + 9363 x^{11} + 37099 x^{10} - 228980 x^{9} - 1521725 x^{8} + 698150 x^{7} + 24552875 x^{6} + 53553625 x^{5} - 77255625 x^{4} - 536596250 x^{3} - 999059375 x^{2} - 858165625 x - 291503125 \)

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:  \(1435834106724053230466538641825103125=5^{5}\cdot 61^{3}\cdot 397^{3}\cdot 5687830021^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $257.32$
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:  $5, 61, 397, 5687830021$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{5} a^{7} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{25} a^{8} - \frac{2}{25} a^{6} + \frac{3}{25} a^{5} + \frac{3}{25} a^{4} - \frac{11}{25} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{125} a^{9} + \frac{3}{125} a^{7} + \frac{53}{125} a^{6} + \frac{18}{125} a^{5} - \frac{46}{125} a^{4} + \frac{2}{5} a^{3} - \frac{12}{25} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{125} a^{10} - \frac{2}{125} a^{8} + \frac{3}{125} a^{7} + \frac{28}{125} a^{6} + \frac{39}{125} a^{5} + \frac{2}{25} a^{4} - \frac{6}{25} a^{3} + \frac{2}{5} a^{2}$, $\frac{1}{625} a^{11} - \frac{2}{625} a^{9} + \frac{3}{625} a^{8} + \frac{3}{625} a^{7} + \frac{289}{625} a^{6} + \frac{37}{125} a^{5} + \frac{29}{125} a^{4} + \frac{4}{25} a^{3} + \frac{11}{25} a^{2} + \frac{1}{5}$, $\frac{1}{3125} a^{12} + \frac{8}{3125} a^{10} + \frac{3}{3125} a^{9} + \frac{33}{3125} a^{8} - \frac{56}{3125} a^{7} - \frac{177}{625} a^{6} + \frac{162}{625} a^{5} + \frac{44}{125} a^{4} + \frac{7}{125} a^{3} + \frac{11}{25} a^{2} - \frac{1}{25} a + \frac{2}{5}$, $\frac{1}{15625} a^{13} - \frac{12}{15625} a^{11} + \frac{53}{15625} a^{10} - \frac{2}{15625} a^{9} + \frac{284}{15625} a^{8} + \frac{296}{3125} a^{7} + \frac{166}{3125} a^{6} - \frac{44}{125} a^{5} + \frac{34}{625} a^{4} + \frac{19}{125} a^{3} - \frac{19}{125} a^{2} + \frac{2}{25} a - \frac{6}{25}$, $\frac{1}{15625} a^{14} - \frac{2}{15625} a^{12} + \frac{3}{15625} a^{11} - \frac{47}{15625} a^{10} + \frac{39}{15625} a^{9} + \frac{7}{3125} a^{8} - \frac{276}{3125} a^{7} - \frac{62}{625} a^{6} - \frac{77}{625} a^{5} + \frac{7}{125} a^{4} - \frac{2}{25} a^{2} + \frac{2}{25} a$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 6692499501300 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T83:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 58320
The 72 conjugacy class representatives for [3^5:2]S(5) are not computed
Character table for [3^5:2]S(5) is not computed

Intermediate fields

5.5.24217.1

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 30 siblings: data not computed
Degree 45 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 ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }$ ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }$ R ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.9.0.1}{9} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{5}$

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
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$61$61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$
61.6.3.1$x^{6} - 122 x^{4} + 3721 x^{2} - 22698100$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
397Data not computed
5687830021Data not computed