↑

Properties

Label 15.9.38583234404...0000.1
Degree $15$
Signature $[9, 3]$
Discriminant $-\,2^{18}\cdot 3^{6}\cdot 5^{6}\cdot 79^{2}\cdot 107^{5}\cdot 1999^{4}\cdot 4987^{2}\cdot 609683^{2}$
Root discriminant $8049.33$
Ramified primes $2, 3, 5, 79, 107, 1999, 4987, 609683$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T97

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4192206848000, 11004542976000, -12458714726400, 7937354137600, -3112713584640, 773045872640, -121534297088, 12280855424, -959105664, 82904760, -6052864, 239912, -13536, 846, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 + 846*x^13 - 13536*x^12 + 239912*x^11 - 6052864*x^10 + 82904760*x^9 - 959105664*x^8 + 12280855424*x^7 - 121534297088*x^6 + 773045872640*x^5 - 3112713584640*x^4 + 7937354137600*x^3 - 12458714726400*x^2 + 11004542976000*x - 4192206848000)
 
gp: K = bnfinit(x^15 + 846*x^13 - 13536*x^12 + 239912*x^11 - 6052864*x^10 + 82904760*x^9 - 959105664*x^8 + 12280855424*x^7 - 121534297088*x^6 + 773045872640*x^5 - 3112713584640*x^4 + 7937354137600*x^3 - 12458714726400*x^2 + 11004542976000*x - 4192206848000, 1)
 

Normalized defining polynomial

\( x^{15} + 846 x^{13} - 13536 x^{12} + 239912 x^{11} - 6052864 x^{10} + 82904760 x^{9} - 959105664 x^{8} + 12280855424 x^{7} - 121534297088 x^{6} + 773045872640 x^{5} - 3112713584640 x^{4} + 7937354137600 x^{3} - 12458714726400 x^{2} + 11004542976000 x - 4192206848000 \)

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:  \(-38583234404050627753560028547647715274328642905059328000000=-\,2^{18}\cdot 3^{6}\cdot 5^{6}\cdot 79^{2}\cdot 107^{5}\cdot 1999^{4}\cdot 4987^{2}\cdot 609683^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $8049.33$
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:  $2, 3, 5, 79, 107, 1999, 4987, 609683$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{20} a^{5} + \frac{1}{10} a^{3} - \frac{1}{10} a^{2}$, $\frac{1}{40} a^{6} + \frac{1}{20} a^{4} + \frac{1}{5} a^{3}$, $\frac{1}{160} a^{7} - \frac{1}{80} a^{5} + \frac{1}{20} a^{4} - \frac{1}{20} a^{3} - \frac{1}{5} a^{2} - \frac{1}{4} a$, $\frac{1}{3200} a^{8} + \frac{7}{1600} a^{6} - \frac{1}{50} a^{5} + \frac{13}{400} a^{4} - \frac{6}{25} a^{3} + \frac{23}{400} a^{2} + \frac{1}{5}$, $\frac{1}{102400} a^{9} - \frac{73}{51200} a^{7} + \frac{33}{3200} a^{6} - \frac{67}{12800} a^{5} - \frac{9}{200} a^{4} + \frac{1463}{12800} a^{3} + \frac{31}{160} a^{2} - \frac{79}{160} a + \frac{1}{4}$, $\frac{1}{409600} a^{10} - \frac{9}{204800} a^{8} + \frac{33}{12800} a^{7} + \frac{157}{51200} a^{6} - \frac{1}{160} a^{5} - \frac{3273}{51200} a^{4} - \frac{453}{3200} a^{3} - \frac{371}{3200} a^{2} + \frac{5}{16} a + \frac{1}{5}$, $\frac{1}{65536000} a^{11} + \frac{1}{1638400} a^{10} - \frac{17}{32768000} a^{9} - \frac{171}{4096000} a^{8} + \frac{13409}{8192000} a^{7} - \frac{5631}{1024000} a^{6} + \frac{21883}{1638400} a^{5} + \frac{68489}{1024000} a^{4} + \frac{34439}{256000} a^{3} - \frac{607}{4000} a^{2} + \frac{427}{6400} a + \frac{351}{800}$, $\frac{1}{1310720000} a^{12} + \frac{463}{655360000} a^{10} - \frac{123}{40960000} a^{9} + \frac{849}{163840000} a^{8} - \frac{6149}{5120000} a^{7} - \frac{71877}{32768000} a^{6} - \frac{95453}{10240000} a^{5} - \frac{292771}{5120000} a^{4} - \frac{45921}{640000} a^{3} + \frac{16939}{128000} a^{2} + \frac{1749}{4000} a - \frac{99}{400}$, $\frac{1}{2621440000000} a^{13} + \frac{3}{16384000000} a^{12} + \frac{3943}{1310720000000} a^{11} - \frac{7533}{81920000000} a^{10} + \frac{989509}{327680000000} a^{9} + \frac{134101}{10240000000} a^{8} - \frac{14755157}{65536000000} a^{7} + \frac{226884717}{20480000000} a^{6} - \frac{282033057}{20480000000} a^{5} - \frac{75924147}{2560000000} a^{4} + \frac{41526349}{256000000} a^{3} - \frac{838157}{8000000} a^{2} + \frac{441601}{3200000} a - \frac{93863}{400000}$, $\frac{1}{2684354560000000} a^{14} + \frac{43}{335544320000000} a^{13} - \frac{464697}{1342177280000000} a^{12} - \frac{752097}{167772160000000} a^{11} - \frac{322246539}{335544320000000} a^{10} + \frac{188637751}{41943040000000} a^{9} + \frac{23810928663}{335544320000000} a^{8} + \frac{5152409779}{41943040000000} a^{7} - \frac{180457712069}{20971520000000} a^{6} - \frac{12817522857}{655360000000} a^{5} + \frac{56166497741}{1310720000000} a^{4} + \frac{849480739}{32768000000} a^{3} + \frac{2231595209}{16384000000} a^{2} - \frac{3869973}{40960000} a + \frac{24501921}{51200000}$

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:  \( 1318066696200000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T97:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 2592000
The 70 conjugacy class representatives for [A(5)^3:2]S(3) are not computed
Character table for [A(5)^3:2]S(3) is not computed

Intermediate fields

3.3.321.1

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 R R ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.9.0.1}{9} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ ${\href{/LocalNumberField/37.9.0.1}{9} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$$[\ ]^{3}$
2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$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$$[\ ]_{2}^{3}$
$79$$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
79.5.0.1$x^{5} - x + 16$$1$$5$$0$$C_5$$[\ ]^{5}$
$107$$\Q_{107}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{107}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{107}$$x + 3$$1$$1$$0$Trivial$[\ ]$
107.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
107.4.2.1$x^{4} + 963 x^{2} + 286225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
107.6.3.1$x^{6} - 214 x^{4} + 11449 x^{2} - 99228483$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
1999Data not computed
4987Data not computed
609683Data not computed