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Properties

Label 15.1.31291515973...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{18}\cdot 5^{7}\cdot 7^{9}\cdot 101^{2}\cdot 6092363^{2}$
Root discriminant $232.47$
Ramified primes $2, 5, 7, 101, 6092363$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T100

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-56000, 392000, -1089200, 1294300, -236880, -699720, 213276, 94472, -28224, 8631, -4876, 357, -216, 9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 + 9*x^13 - 216*x^12 + 357*x^11 - 4876*x^10 + 8631*x^9 - 28224*x^8 + 94472*x^7 + 213276*x^6 - 699720*x^5 - 236880*x^4 + 1294300*x^3 - 1089200*x^2 + 392000*x - 56000)
 
gp: K = bnfinit(x^15 + 9*x^13 - 216*x^12 + 357*x^11 - 4876*x^10 + 8631*x^9 - 28224*x^8 + 94472*x^7 + 213276*x^6 - 699720*x^5 - 236880*x^4 + 1294300*x^3 - 1089200*x^2 + 392000*x - 56000, 1)
 

Normalized defining polynomial

\( x^{15} + 9 x^{13} - 216 x^{12} + 357 x^{11} - 4876 x^{10} + 8631 x^{9} - 28224 x^{8} + 94472 x^{7} + 213276 x^{6} - 699720 x^{5} - 236880 x^{4} + 1294300 x^{3} - 1089200 x^{2} + 392000 x - 56000 \)

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:  \(-312915159730527430614453923840000000=-\,2^{18}\cdot 5^{7}\cdot 7^{9}\cdot 101^{2}\cdot 6092363^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $232.47$
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:  $2, 5, 7, 101, 6092363$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{10} a^{9} - \frac{2}{5} a^{8} + \frac{1}{10} a^{7} - \frac{2}{5} a^{6} - \frac{1}{10} a^{5} + \frac{2}{5} a^{4} - \frac{1}{10} a^{3} + \frac{2}{5} a^{2}$, $\frac{1}{20} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{3}{20} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{3}{10} a^{2} - \frac{1}{2} a$, $\frac{1}{20} a^{11} - \frac{1}{20} a^{9} - \frac{3}{10} a^{8} + \frac{7}{20} a^{7} - \frac{3}{10} a^{6} + \frac{1}{20} a^{5} + \frac{3}{10} a^{4} + \frac{1}{10} a^{3} + \frac{3}{10} a^{2}$, $\frac{1}{200} a^{12} - \frac{1}{200} a^{10} - \frac{3}{100} a^{9} + \frac{47}{200} a^{8} + \frac{7}{100} a^{7} - \frac{59}{200} a^{6} - \frac{27}{100} a^{5} + \frac{41}{100} a^{4} - \frac{7}{100} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{1000} a^{13} + \frac{19}{1000} a^{11} - \frac{2}{125} a^{10} + \frac{47}{1000} a^{9} + \frac{58}{125} a^{8} - \frac{399}{1000} a^{7} - \frac{121}{250} a^{6} - \frac{9}{500} a^{5} - \frac{8}{125} a^{4} - \frac{2}{5} a^{3} + \frac{9}{50} a^{2} + \frac{3}{10} a - \frac{2}{5}$, $\frac{1}{281604768356534755215220287130000} a^{14} + \frac{412698294293209334230593421}{5632095367130695104304405742600} a^{13} + \frac{664544127978821477670420202919}{281604768356534755215220287130000} a^{12} - \frac{1852665974270478288872333053433}{140802384178267377607610143565000} a^{11} + \frac{247384466066437140242557423847}{281604768356534755215220287130000} a^{10} + \frac{6373402652705041560412974428507}{140802384178267377607610143565000} a^{9} - \frac{9828700865539745283996673941799}{281604768356534755215220287130000} a^{8} - \frac{5885093823560754917874152126817}{140802384178267377607610143565000} a^{7} - \frac{8263604656697861266742894288109}{140802384178267377607610143565000} a^{6} + \frac{34727442887492920105661575909709}{70401192089133688803805071782500} a^{5} + \frac{485811004791937021229698036361}{2816047683565347552152202871300} a^{4} + \frac{2014918508871110560269605610109}{14080238417826737760761014356500} a^{3} + \frac{1144244202531351287177567235657}{2816047683565347552152202871300} a^{2} + \frac{388266245452574530402473756603}{1408023841782673776076101435650} a - \frac{40662513565298648494849984177}{140802384178267377607610143565}$

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:  \( 13462022613800 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T100:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 5184000
The 79 conjugacy class representatives for [1/2.S(5)^3]S(3) are not computed
Character table for [1/2.S(5)^3]S(3) is not computed

Intermediate fields

3.1.140.1

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 $15$ R R ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ $15$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ $15$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.6.8$x^{6} + 2 x + 2$$6$$1$$6$$S_4$$[4/3, 4/3]_{3}^{2}$
2.6.10.4$x^{6} + 2 x^{5} + 2 x^{4} + 6$$6$$1$$10$$S_4$$[8/3, 8/3]_{3}^{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.10.9.1$x^{10} - 7$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
101Data not computed
6092363Data not computed