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Properties

Label 15.9.48312643100...0000.1
Degree $15$
Signature $[9, 3]$
Discriminant $-\,2^{15}\cdot 5^{6}\cdot 7^{14}\cdot 113^{4}\cdot 292113919^{2}$
Root discriminant $1110.72$
Ramified primes $2, 5, 7, 113, 292113919$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T101

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3239936000, 17009664000, 38514739200, 49074905600, 38490439680, 19089311360, 5909943424, 1073983904, 91965888, -2073960, -1126048, -72268, -1008, -126, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 126*x^13 - 1008*x^12 - 72268*x^11 - 1126048*x^10 - 2073960*x^9 + 91965888*x^8 + 1073983904*x^7 + 5909943424*x^6 + 19089311360*x^5 + 38490439680*x^4 + 49074905600*x^3 + 38514739200*x^2 + 17009664000*x + 3239936000)
 
gp: K = bnfinit(x^15 - 126*x^13 - 1008*x^12 - 72268*x^11 - 1126048*x^10 - 2073960*x^9 + 91965888*x^8 + 1073983904*x^7 + 5909943424*x^6 + 19089311360*x^5 + 38490439680*x^4 + 49074905600*x^3 + 38514739200*x^2 + 17009664000*x + 3239936000, 1)
 

Normalized defining polynomial

\( x^{15} - 126 x^{13} - 1008 x^{12} - 72268 x^{11} - 1126048 x^{10} - 2073960 x^{9} + 91965888 x^{8} + 1073983904 x^{7} + 5909943424 x^{6} + 19089311360 x^{5} + 38490439680 x^{4} + 49074905600 x^{3} + 38514739200 x^{2} + 17009664000 x + 3239936000 \)

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:  \(-4831264310017778731576326598323772072448000000=-\,2^{15}\cdot 5^{6}\cdot 7^{14}\cdot 113^{4}\cdot 292113919^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1110.72$
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:  $2, 5, 7, 113, 292113919$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{40} a^{5} - \frac{1}{20} a^{3} + \frac{1}{10} a^{2}$, $\frac{1}{160} a^{6} - \frac{1}{80} a^{5} + \frac{1}{20} a^{4} + \frac{1}{20} a^{3} + \frac{3}{40} a^{2} - \frac{1}{4} a$, $\frac{1}{160} a^{7} + \frac{1}{40} a^{4} - \frac{1}{40} a^{3} + \frac{1}{20} a^{2}$, $\frac{1}{1600} a^{8} - \frac{1}{400} a^{6} + \frac{1}{200} a^{5} + \frac{21}{400} a^{4} + \frac{9}{100} a^{3} + \frac{4}{25} a^{2} + \frac{2}{5}$, $\frac{1}{6400} a^{9} - \frac{7}{3200} a^{7} + \frac{1}{800} a^{6} + \frac{1}{1600} a^{5} - \frac{3}{200} a^{4} + \frac{7}{800} a^{3} - \frac{1}{8} a^{2} - \frac{1}{40} a - \frac{1}{2}$, $\frac{1}{25600} a^{10} - \frac{1}{12800} a^{9} - \frac{3}{12800} a^{8} - \frac{1}{6400} a^{7} - \frac{11}{6400} a^{6} - \frac{1}{128} a^{5} + \frac{39}{640} a^{4} - \frac{11}{320} a^{3} + \frac{29}{800} a^{2} - \frac{39}{80} a + \frac{9}{20}$, $\frac{1}{512000} a^{11} - \frac{13}{256000} a^{9} + \frac{7}{32000} a^{8} + \frac{243}{128000} a^{7} + \frac{11}{16000} a^{6} + \frac{101}{12800} a^{5} + \frac{87}{8000} a^{4} - \frac{469}{8000} a^{3} + \frac{933}{4000} a^{2} + \frac{363}{800} a - \frac{59}{200}$, $\frac{1}{10240000} a^{12} - \frac{1}{1024000} a^{11} - \frac{53}{5120000} a^{10} - \frac{187}{2560000} a^{9} + \frac{723}{2560000} a^{8} - \frac{3691}{1280000} a^{7} + \frac{229}{256000} a^{6} - \frac{4217}{640000} a^{5} + \frac{4791}{160000} a^{4} + \frac{1287}{20000} a^{3} - \frac{31}{128} a^{2} + \frac{207}{8000} a - \frac{33}{400}$, $\frac{1}{10240000000} a^{13} - \frac{3}{128000000} a^{12} + \frac{817}{5120000000} a^{11} - \frac{5723}{640000000} a^{10} + \frac{15373}{2560000000} a^{9} - \frac{16469}{320000000} a^{8} + \frac{357367}{256000000} a^{7} + \frac{390147}{160000000} a^{6} - \frac{1000163}{320000000} a^{5} + \frac{3708223}{80000000} a^{4} - \frac{1335059}{16000000} a^{3} + \frac{62219}{500000} a^{2} + \frac{66609}{800000} a + \frac{83367}{200000}$, $\frac{1}{2621440000000} a^{14} - \frac{11}{655360000000} a^{13} + \frac{60297}{1310720000000} a^{12} - \frac{258913}{327680000000} a^{11} - \frac{8070459}{655360000000} a^{10} + \frac{4999839}{163840000000} a^{9} - \frac{88875861}{327680000000} a^{8} - \frac{116481291}{81920000000} a^{7} - \frac{75985039}{81920000000} a^{6} - \frac{11751257}{10240000000} a^{5} + \frac{304549413}{20480000000} a^{4} + \frac{29922261}{1024000000} a^{3} + \frac{22463937}{1024000000} a^{2} + \frac{126129}{25600000} a + \frac{5842233}{12800000}$

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:  \( 432308459235000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T101:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 5184000
The 133 conjugacy class representatives for [S(5)^3]3=S(5)wr3 are not computed
Character table for [S(5)^3]3=S(5)wr3 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\)

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R R ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.9.0.1}{9} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.9.0.1}{9} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.9.0.1}{9} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.9.0.1}{9} }{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$$[\ ]^{3}$
2.6.9.8$x^{6} + 4 x^{2} - 24$$2$$3$$9$$A_4\times C_2$$[2, 2, 3]^{3}$
2.6.6.2$x^{6} - x^{4} - 5$$2$$3$$6$$A_4\times C_2$$[2, 2]^{6}$
$5$5.3.0.1$x^{3} - x + 2$$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}$
7Data not computed
113Data not computed
292113919Data not computed