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Properties

Label 15.1.91288227819...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{18}\cdot 3^{13}\cdot 5^{17}\cdot 31^{5}$
Root discriminant $115.89$
Ramified primes $2, 3, 5, 31$
Class number $1$ (GRH)
Class group Trivial (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![-2381172, -4945740, -3816360, -1831320, -403920, 420532, 238660, 4440, -26970, -8220, 561, 765, 60, -30, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 30*x^13 + 60*x^12 + 765*x^11 + 561*x^10 - 8220*x^9 - 26970*x^8 + 4440*x^7 + 238660*x^6 + 420532*x^5 - 403920*x^4 - 1831320*x^3 - 3816360*x^2 - 4945740*x - 2381172)
 
gp: K = bnfinit(x^15 - 5*x^14 - 30*x^13 + 60*x^12 + 765*x^11 + 561*x^10 - 8220*x^9 - 26970*x^8 + 4440*x^7 + 238660*x^6 + 420532*x^5 - 403920*x^4 - 1831320*x^3 - 3816360*x^2 - 4945740*x - 2381172, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} - 30 x^{13} + 60 x^{12} + 765 x^{11} + 561 x^{10} - 8220 x^{9} - 26970 x^{8} + 4440 x^{7} + 238660 x^{6} + 420532 x^{5} - 403920 x^{4} - 1831320 x^{3} - 3816360 x^{2} - 4945740 x - 2381172 \)

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:  \(-9128822781954600000000000000000=-\,2^{18}\cdot 3^{13}\cdot 5^{17}\cdot 31^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $115.89$
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:  $2, 3, 5, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} + \frac{2}{5}$, $\frac{1}{5} a^{6} + \frac{2}{5} a$, $\frac{1}{5} a^{7} + \frac{2}{5} a^{2}$, $\frac{1}{5} a^{8} + \frac{2}{5} a^{3}$, $\frac{1}{25} a^{9} - \frac{2}{25} a^{8} - \frac{1}{25} a^{7} + \frac{2}{25} a^{6} + \frac{1}{25} a^{5} + \frac{2}{25} a^{4} - \frac{4}{25} a^{3} - \frac{2}{25} a^{2} + \frac{4}{25} a + \frac{2}{25}$, $\frac{1}{150} a^{10} - \frac{1}{150} a^{9} - \frac{4}{75} a^{8} - \frac{7}{75} a^{7} - \frac{7}{150} a^{6} + \frac{13}{150} a^{5} + \frac{8}{25} a^{4} - \frac{11}{25} a^{3} + \frac{12}{25} a^{2} + \frac{6}{25} a + \frac{12}{25}$, $\frac{1}{150} a^{11} - \frac{1}{50} a^{9} - \frac{2}{75} a^{8} + \frac{1}{50} a^{7} - \frac{2}{25} a^{6} + \frac{7}{150} a^{5} - \frac{1}{25} a^{4} + \frac{7}{25} a^{3} + \frac{1}{25} a^{2} + \frac{12}{25} a - \frac{6}{25}$, $\frac{1}{750} a^{12} - \frac{1}{750} a^{11} - \frac{1}{750} a^{10} - \frac{1}{50} a^{9} - \frac{3}{50} a^{8} - \frac{31}{750} a^{7} + \frac{41}{750} a^{6} + \frac{31}{750} a^{5} - \frac{1}{25} a^{4} - \frac{3}{25} a^{3} - \frac{11}{125} a^{2} + \frac{56}{125} a + \frac{11}{125}$, $\frac{1}{750} a^{13} - \frac{1}{375} a^{11} - \frac{1}{750} a^{10} - \frac{1}{50} a^{9} - \frac{8}{375} a^{8} + \frac{4}{75} a^{7} - \frac{21}{250} a^{6} - \frac{22}{375} a^{5} - \frac{1}{25} a^{4} - \frac{6}{125} a^{3} + \frac{11}{25} a^{2} + \frac{22}{125} a - \frac{14}{125}$, $\frac{1}{247959947885404481842096084500} a^{14} - \frac{10849863800276854223554829}{16530663192360298789473072300} a^{13} + \frac{23861261928332977559929957}{41326657980900746973682680750} a^{12} - \frac{3582790839769670793023663}{8265331596180149394736536150} a^{11} + \frac{68673738539834859632853157}{82653315961801493947365361500} a^{10} + \frac{356198499268823773150812891}{27551105320600497982455120500} a^{9} - \frac{789289152447648824972645359}{8265331596180149394736536150} a^{8} - \frac{858772660121720051536567081}{20663328990450373486841340375} a^{7} - \frac{258077283841307317378866947}{2755110532060049798245512050} a^{6} - \frac{1131644676267496318766713019}{61989986971351120460524021125} a^{5} - \frac{5570278072140306777199142887}{13775552660300248991227560250} a^{4} + \frac{100621517647598524665830583}{275511053206004979824551205} a^{3} + \frac{1294037110128233564630596683}{6887776330150124495613780125} a^{2} - \frac{1327095665329267591639731}{8887453329225967091114555} a + \frac{300220870575639643472220637}{666558999691947531833591625}$

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:  $7$
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:  \( 24782775558.00696 \) (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

3.1.9300.1, 5.1.4050000.4

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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$$[\ ]_{5}^{4}$
2.10.14.5$x^{10} - 2 x^{5} - 2$$10$$1$$14$$F_{5}\times C_2$$[2]_{5}^{4}$
$3$3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.10.9.2$x^{10} + 3$$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$$[5/4]_{12}^{2}$
$31$$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$