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Properties

Label 15.1.54777026874...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{23}\cdot 3^{12}\cdot 5^{17}\cdot 11^{5}$
Root discriminant $96.07$
Ramified primes $2, 3, 5, 11$
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![-250228, -319000, -150700, -120600, -78250, 63307, 9000, -22675, -8400, -250, 964, 250, -50, -25, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 25*x^13 - 50*x^12 + 250*x^11 + 964*x^10 - 250*x^9 - 8400*x^8 - 22675*x^7 + 9000*x^6 + 63307*x^5 - 78250*x^4 - 120600*x^3 - 150700*x^2 - 319000*x - 250228)
 
gp: K = bnfinit(x^15 - 25*x^13 - 50*x^12 + 250*x^11 + 964*x^10 - 250*x^9 - 8400*x^8 - 22675*x^7 + 9000*x^6 + 63307*x^5 - 78250*x^4 - 120600*x^3 - 150700*x^2 - 319000*x - 250228, 1)
 

Normalized defining polynomial

\( x^{15} - 25 x^{13} - 50 x^{12} + 250 x^{11} + 964 x^{10} - 250 x^{9} - 8400 x^{8} - 22675 x^{7} + 9000 x^{6} + 63307 x^{5} - 78250 x^{4} - 120600 x^{3} - 150700 x^{2} - 319000 x - 250228 \)

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:  \(-547770268742400000000000000000=-\,2^{23}\cdot 3^{12}\cdot 5^{17}\cdot 11^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $96.07$
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:  $2, 3, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} - \frac{2}{5}$, $\frac{1}{10} a^{6} - \frac{1}{10} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{10} a^{7} - \frac{1}{2} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{10} a^{8} - \frac{1}{2} a^{4} - \frac{1}{5} a^{3}$, $\frac{1}{50} a^{9} + \frac{1}{25} a^{8} - \frac{1}{50} a^{7} - \frac{1}{25} a^{6} + \frac{1}{50} a^{5} - \frac{6}{25} a^{4} + \frac{1}{50} a^{3} + \frac{6}{25} a^{2} + \frac{12}{25} a - \frac{6}{25}$, $\frac{1}{100} a^{10} - \frac{1}{100} a^{9} - \frac{1}{50} a^{8} - \frac{1}{25} a^{7} - \frac{3}{100} a^{6} - \frac{1}{20} a^{5} + \frac{3}{25} a^{4} - \frac{13}{50} a^{3} + \frac{12}{25} a^{2} + \frac{9}{25} a + \frac{4}{25}$, $\frac{1}{100} a^{11} - \frac{1}{100} a^{9} - \frac{1}{50} a^{8} + \frac{1}{100} a^{7} - \frac{1}{50} a^{6} - \frac{1}{100} a^{5} + \frac{3}{25} a^{4} + \frac{6}{25} a^{3} - \frac{3}{25} a^{2} - \frac{1}{5} a + \frac{3}{25}$, $\frac{1}{500} a^{12} + \frac{1}{500} a^{11} - \frac{1}{500} a^{10} - \frac{1}{100} a^{9} + \frac{3}{100} a^{8} - \frac{19}{500} a^{7} + \frac{11}{500} a^{6} + \frac{19}{500} a^{5} - \frac{19}{50} a^{4} - \frac{9}{25} a^{3} + \frac{21}{125} a^{2} + \frac{56}{125} a - \frac{21}{125}$, $\frac{1}{11000} a^{13} - \frac{1}{5500} a^{12} - \frac{27}{5500} a^{11} - \frac{13}{2750} a^{10} - \frac{1}{550} a^{9} + \frac{9}{2750} a^{8} + \frac{59}{5500} a^{7} - \frac{91}{2750} a^{6} + \frac{603}{11000} a^{5} + \frac{49}{1100} a^{4} + \frac{678}{1375} a^{3} + \frac{234}{1375} a^{2} + \frac{13}{125} a - \frac{17}{250}$, $\frac{1}{487230555608851325909000} a^{14} + \frac{12337760459071103343}{487230555608851325909000} a^{13} - \frac{83111543836032774851}{243615277804425662954500} a^{12} - \frac{94129261918312643567}{48723055560885132590900} a^{11} - \frac{59961668796555048791}{22146843436765969359500} a^{10} + \frac{391081738652979448687}{60903819451106415738625} a^{9} + \frac{3047134338140132779777}{121807638902212831477250} a^{8} + \frac{1806086033240206270889}{243615277804425662954500} a^{7} + \frac{828495575673637631253}{97446111121770265181800} a^{6} - \frac{4400658993868236786137}{44293686873531938719000} a^{5} + \frac{62740951633898962384327}{243615277804425662954500} a^{4} - \frac{29323119857192425980547}{121807638902212831477250} a^{3} + \frac{3085795678633186313969}{60903819451106415738625} a^{2} - \frac{187118347628863264007}{442936868735319387190} a + \frac{4837841838613335828363}{11073421718382984679750}$

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:  \( 7825223971.25853 \) (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

3.1.2200.1, 5.1.4050000.3

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ R ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$[\ ]_{5}^{4}$
2.10.19.17$x^{10} - 2 x^{4} + 4 x^{2} - 10$$10$$1$$19$$F_{5}\times C_2$$[3]_{5}^{4}$
$3$3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.10.8.1$x^{10} - 3 x^{5} + 18$$5$$2$$8$$F_5$$[\ ]_{5}^{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$$[5/4]_{12}^{2}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$