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Properties

Label 15.1.92375919403...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{10}\cdot 5^{27}\cdot 11^{12}\cdot 131^{5}$
Root discriminant $994.73$
Ramified primes $2, 5, 11, 131$
Class number $200$ (GRH)
Class group $[5, 40]$ (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![325852261250, 6382037500, -3290217500, -2382376375, -219639400, 145166318, -5716705, 3169175, -175530, -66945, 19844, 285, -95, 25, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 + 25*x^13 - 95*x^12 + 285*x^11 + 19844*x^10 - 66945*x^9 - 175530*x^8 + 3169175*x^7 - 5716705*x^6 + 145166318*x^5 - 219639400*x^4 - 2382376375*x^3 - 3290217500*x^2 + 6382037500*x + 325852261250)
 
gp: K = bnfinit(x^15 - 5*x^14 + 25*x^13 - 95*x^12 + 285*x^11 + 19844*x^10 - 66945*x^9 - 175530*x^8 + 3169175*x^7 - 5716705*x^6 + 145166318*x^5 - 219639400*x^4 - 2382376375*x^3 - 3290217500*x^2 + 6382037500*x + 325852261250, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} + 25 x^{13} - 95 x^{12} + 285 x^{11} + 19844 x^{10} - 66945 x^{9} - 175530 x^{8} + 3169175 x^{7} - 5716705 x^{6} + 145166318 x^{5} - 219639400 x^{4} - 2382376375 x^{3} - 3290217500 x^{2} + 6382037500 x + 325852261250 \)

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:  \(-923759194031620398058860778808593750000000000=-\,2^{10}\cdot 5^{27}\cdot 11^{12}\cdot 131^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $994.73$
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:  $2, 5, 11, 131$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{7} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2}$, $\frac{1}{5} a^{8} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2}$, $\frac{1}{25} a^{9} + \frac{2}{25} a^{8} + \frac{2}{25} a^{7} + \frac{1}{25} a^{6} + \frac{11}{25} a^{5} - \frac{12}{25} a^{4} - \frac{8}{25} a^{3} - \frac{2}{5} a^{2}$, $\frac{1}{25} a^{10} - \frac{2}{25} a^{8} + \frac{2}{25} a^{7} - \frac{1}{25} a^{6} + \frac{1}{25} a^{5} + \frac{11}{25} a^{4} + \frac{1}{25} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{25} a^{11} + \frac{1}{25} a^{8} - \frac{2}{25} a^{7} - \frac{2}{25} a^{6} + \frac{8}{25} a^{5} - \frac{3}{25} a^{4} - \frac{6}{25} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{125} a^{12} + \frac{1}{125} a^{11} - \frac{2}{125} a^{10} - \frac{9}{125} a^{8} + \frac{2}{25} a^{7} + \frac{12}{125} a^{6} + \frac{27}{125} a^{5} + \frac{41}{125} a^{4} + \frac{11}{25} a^{3} + \frac{2}{5} a^{2}$, $\frac{1}{125} a^{13} + \frac{2}{125} a^{11} + \frac{2}{125} a^{10} + \frac{1}{125} a^{9} - \frac{6}{125} a^{8} + \frac{12}{125} a^{7} - \frac{2}{25} a^{6} + \frac{14}{125} a^{5} + \frac{29}{125} a^{4} - \frac{3}{25} a^{3}$, $\frac{1}{42431312162236939677704937517205877332649534401716989187625} a^{14} + \frac{45543409764369933234664737655097392564613507912836050074}{42431312162236939677704937517205877332649534401716989187625} a^{13} + \frac{104246719093861370832170765497943494545687876392880663864}{42431312162236939677704937517205877332649534401716989187625} a^{12} + \frac{412544625413501494264872461890691164626631050653356935952}{42431312162236939677704937517205877332649534401716989187625} a^{11} + \frac{31727472280540713057856910728073542170610784077277578346}{8486262432447387935540987503441175466529906880343397837525} a^{10} + \frac{42602852783653943799356629095218214511167339983816982533}{42431312162236939677704937517205877332649534401716989187625} a^{9} - \frac{82568464194538044645355987400080947389045903670090581641}{8486262432447387935540987503441175466529906880343397837525} a^{8} - \frac{988520172754298936864839850910934388032483336112293028942}{42431312162236939677704937517205877332649534401716989187625} a^{7} - \frac{2927732601085726932390646378054510831868414041104062545352}{42431312162236939677704937517205877332649534401716989187625} a^{6} - \frac{18051580731895105108456799461695012099327758225720762120746}{42431312162236939677704937517205877332649534401716989187625} a^{5} + \frac{20465828201433919922926744937659834884845279071146737052478}{42431312162236939677704937517205877332649534401716989187625} a^{4} - \frac{2459916151795751664147702295423844328262760269647338184746}{8486262432447387935540987503441175466529906880343397837525} a^{3} + \frac{109434948177394623630579708659579631887043357516032949474}{339450497297895517421639500137647018661196275213735913501} a^{2} + \frac{106692437026860566972914081060768917657909090420491753887}{339450497297895517421639500137647018661196275213735913501} a - \frac{147798436322214798824364599314818794437221448588085849722}{339450497297895517421639500137647018661196275213735913501}$

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

Class group and class number

$C_{5}\times C_{40}$, which has order $200$ (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:  \( 411098018557015.94 \) (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

3.1.524.1, 5.1.28595703125.1

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 ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ R ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.12.8.1$x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
$5$5.5.9.3$x^{5} + 80$$5$$1$$9$$F_5$$[9/4]_{4}$
5.5.9.3$x^{5} + 80$$5$$1$$9$$F_5$$[9/4]_{4}$
5.5.9.3$x^{5} + 80$$5$$1$$9$$F_5$$[9/4]_{4}$
$11$11.5.4.5$x^{5} - 99$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.5$x^{5} - 99$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.5$x^{5} - 99$$5$$1$$4$$C_5$$[\ ]_{5}$
$131$$\Q_{131}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{131}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{131}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{131}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{131}$$x + 3$$1$$1$$0$Trivial$[\ ]$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$