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Properties

Label 15.9.29492759518...0000.1
Degree $15$
Signature $[9, 3]$
Discriminant $-\,2^{6}\cdot 5^{6}\cdot 7^{10}\cdot 13\cdot 197^{4}\cdot 6791^{2}\cdot 10753^{2}$
Root discriminant $498.86$
Ramified primes $2, 5, 7, 13, 197, 6791, 10753$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T101

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![197000, 2068500, 9367350, 23871475, 37445760, 37151415, 23061222, 8533899, 1674756, 108945, -11774, -1843, -396, -99, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 99*x^13 - 396*x^12 - 1843*x^11 - 11774*x^10 + 108945*x^9 + 1674756*x^8 + 8533899*x^7 + 23061222*x^6 + 37151415*x^5 + 37445760*x^4 + 23871475*x^3 + 9367350*x^2 + 2068500*x + 197000)
 
gp: K = bnfinit(x^15 - 99*x^13 - 396*x^12 - 1843*x^11 - 11774*x^10 + 108945*x^9 + 1674756*x^8 + 8533899*x^7 + 23061222*x^6 + 37151415*x^5 + 37445760*x^4 + 23871475*x^3 + 9367350*x^2 + 2068500*x + 197000, 1)
 

Normalized defining polynomial

\( x^{15} - 99 x^{13} - 396 x^{12} - 1843 x^{11} - 11774 x^{10} + 108945 x^{9} + 1674756 x^{8} + 8533899 x^{7} + 23061222 x^{6} + 37151415 x^{5} + 37445760 x^{4} + 23871475 x^{3} + 9367350 x^{2} + 2068500 x + 197000 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[9, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-29492759518563811794405922500363613000000=-\,2^{6}\cdot 5^{6}\cdot 7^{10}\cdot 13\cdot 197^{4}\cdot 6791^{2}\cdot 10753^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $498.86$
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:  $2, 5, 7, 13, 197, 6791, 10753$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2}$, $\frac{1}{5} a^{6} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3}$, $\frac{1}{5} a^{7} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{25} a^{8} - \frac{1}{25} a^{6} + \frac{1}{25} a^{5} - \frac{1}{25} a^{4} - \frac{8}{25} a^{3} + \frac{4}{25} a^{2} - \frac{2}{5}$, $\frac{1}{25} a^{9} - \frac{1}{25} a^{7} + \frac{1}{25} a^{6} - \frac{1}{25} a^{5} - \frac{8}{25} a^{4} + \frac{4}{25} a^{3} - \frac{2}{5} a$, $\frac{1}{50} a^{10} - \frac{1}{50} a^{9} - \frac{3}{50} a^{7} - \frac{3}{50} a^{6} - \frac{1}{50} a^{5} + \frac{21}{50} a^{4} + \frac{9}{25} a^{3} + \frac{7}{25} a^{2} - \frac{3}{10} a - \frac{1}{5}$, $\frac{1}{250} a^{11} + \frac{1}{250} a^{9} - \frac{1}{250} a^{8} + \frac{11}{125} a^{7} - \frac{2}{125} a^{6} + \frac{111}{250} a^{4} - \frac{18}{125} a^{3} + \frac{67}{250} a^{2} + \frac{21}{50} a - \frac{8}{25}$, $\frac{1}{1250} a^{12} + \frac{3}{625} a^{10} + \frac{2}{625} a^{9} + \frac{6}{625} a^{8} + \frac{21}{1250} a^{7} + \frac{1}{250} a^{6} + \frac{43}{625} a^{5} - \frac{101}{1250} a^{4} + \frac{577}{1250} a^{3} - \frac{33}{250} a^{2} - \frac{51}{250} a + \frac{6}{25}$, $\frac{1}{2500000} a^{13} - \frac{31}{125000} a^{12} + \frac{2841}{2500000} a^{11} - \frac{827}{312500} a^{10} + \frac{22217}{2500000} a^{9} + \frac{1523}{1250000} a^{8} - \frac{15279}{500000} a^{7} - \frac{1172}{78125} a^{6} - \frac{176921}{2500000} a^{5} - \frac{485959}{1250000} a^{4} + \frac{195047}{500000} a^{3} + \frac{54517}{125000} a^{2} + \frac{18703}{100000} a - \frac{2511}{50000}$, $\frac{1}{80000000} a^{14} + \frac{1}{8000000} a^{13} + \frac{10241}{80000000} a^{12} + \frac{76607}{40000000} a^{11} + \frac{542137}{80000000} a^{10} - \frac{384561}{20000000} a^{9} - \frac{50283}{16000000} a^{8} - \frac{184177}{40000000} a^{7} - \frac{2394441}{80000000} a^{6} + \frac{1048963}{20000000} a^{5} - \frac{7312221}{16000000} a^{4} + \frac{291439}{8000000} a^{3} - \frac{931929}{3200000} a^{2} - \frac{306933}{800000} a + \frac{25407}{160000}$

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:  $11$
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:  \( 1141197199090000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T101:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 5184000
The 133 conjugacy class representatives for [S(5)^3]3=S(5)wr3 are not computed
Character table for [S(5)^3]3=S(5)wr3 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 18 sibling: data not computed
Degree 30 siblings: data not computed
Degree 36 siblings: data not computed
Degree 45 sibling: 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 ${\href{/LocalNumberField/17.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/19.9.0.1}{9} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.9.0.1}{9} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ $15$

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$$[\ ]^{3}$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.6.6$x^{6} - 13 x^{4} + 7 x^{2} - 3$$2$$3$$6$$A_4\times C_2$$[2, 2, 2]^{3}$
$5$5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
197Data not computed
6791Data not computed
10753Data not computed