Normalized defining polynomial
\( x^{15} - 1926 x^{13} - 15408 x^{12} + 1171432 x^{11} + 19205152 x^{10} - 128106360 x^{9} - 5496558912 x^{8} - 57384566496 x^{7} - 311196686976 x^{6} - 1003145984640 x^{5} - 2022280888320 x^{4} - 2578386854400 x^{3} - 2023557580800 x^{2} - 893684736000 x - 170225664000 \)
Invariants
| Degree: | $15$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[15, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(11006353820590761829104554853487124581411222177394688000000=2^{18}\cdot 3^{10}\cdot 5^{6}\cdot 7^{4}\cdot 11^{2}\cdot 107^{5}\cdot 239^{2}\cdot 1979^{4}\cdot 112901^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $7403.60$ | 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: | $2, 3, 5, 7, 11, 107, 239, 1979, 112901$ | 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 \(a\))
$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{20} a^{5} - \frac{1}{10} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{40} a^{6} - \frac{1}{20} a^{4} + \frac{1}{10} a^{3}$, $\frac{1}{80} a^{7} - \frac{1}{40} a^{5} + \frac{1}{20} a^{4} - \frac{1}{2} a$, $\frac{1}{800} a^{8} + \frac{3}{400} a^{6} + \frac{1}{100} a^{5} + \frac{2}{25} a^{4} - \frac{1}{50} a^{3} + \frac{7}{100} a^{2} - \frac{1}{5}$, $\frac{1}{3200} a^{9} - \frac{7}{1600} a^{7} - \frac{1}{100} a^{6} - \frac{7}{400} a^{5} + \frac{3}{25} a^{4} + \frac{77}{400} a^{3} - \frac{1}{5} a^{2} - \frac{1}{20} a$, $\frac{1}{6400} a^{10} + \frac{1}{3200} a^{8} - \frac{1}{200} a^{7} + \frac{1}{160} a^{6} - \frac{1}{50} a^{5} + \frac{1}{160} a^{4} - \frac{19}{100} a^{3} + \frac{43}{200} a^{2} - \frac{2}{5}$, $\frac{1}{128000} a^{11} + \frac{7}{64000} a^{9} - \frac{3}{8000} a^{8} + \frac{1}{250} a^{7} - \frac{9}{4000} a^{6} - \frac{67}{3200} a^{5} + \frac{217}{2000} a^{4} + \frac{431}{2000} a^{3} - \frac{87}{1000} a^{2} + \frac{53}{200} a + \frac{1}{50}$, $\frac{1}{1280000} a^{12} + \frac{47}{640000} a^{10} + \frac{3}{20000} a^{9} - \frac{9}{40000} a^{8} - \frac{47}{20000} a^{7} - \frac{323}{32000} a^{6} - \frac{273}{20000} a^{5} + \frac{2161}{20000} a^{4} + \frac{291}{1250} a^{3} - \frac{59}{2000} a^{2} + \frac{93}{250} a + \frac{4}{25}$, $\frac{1}{56320000000} a^{13} - \frac{531}{1408000000} a^{12} - \frac{75083}{28160000000} a^{11} - \frac{38999}{1760000000} a^{10} + \frac{482649}{7040000000} a^{9} + \frac{1083131}{1760000000} a^{8} - \frac{482643}{1408000000} a^{7} - \frac{92737}{20000000} a^{6} + \frac{5707637}{1760000000} a^{5} + \frac{2710743}{40000000} a^{4} - \frac{18102509}{88000000} a^{3} + \frac{395401}{11000000} a^{2} - \frac{1314741}{4400000} a - \frac{265083}{1100000}$, $\frac{1}{7208960000000} a^{14} - \frac{1}{163840000000} a^{13} + \frac{59397}{3604480000000} a^{12} - \frac{3450813}{901120000000} a^{11} + \frac{20237433}{901120000000} a^{10} + \frac{13533591}{112640000000} a^{9} - \frac{50271501}{81920000000} a^{8} - \frac{829195141}{225280000000} a^{7} - \frac{397306139}{225280000000} a^{6} - \frac{44929091}{14080000000} a^{5} - \frac{4181361637}{56320000000} a^{4} - \frac{49893599}{256000000} a^{3} + \frac{213487487}{2816000000} a^{2} + \frac{34286479}{70400000} a - \frac{14285817}{35200000}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $14$ | 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: | \( 4636547341530000000000000 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| 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 | R | R | $15$ | ${\href{/LocalNumberField/17.9.0.1}{9} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }$ | ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ | ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ | ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ | ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ | ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ | ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ | ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ | ${\href{/LocalNumberField/59.5.0.1}{5} }^{3}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| $2$ | 2.3.0.1 | $x^{3} - x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 2.6.9.8 | $x^{6} + 4 x^{2} - 24$ | $2$ | $3$ | $9$ | $A_4\times C_2$ | $[2, 2, 3]^{3}$ | |
| 2.6.9.2 | $x^{6} + 4 x^{2} - 8$ | $2$ | $3$ | $9$ | $A_4\times C_2$ | $[2, 2, 3]^{3}$ | |
| $3$ | 3.2.1.1 | $x^{2} - 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.3.0.1 | $x^{3} - x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 3.10.9.2 | $x^{10} + 3$ | $10$ | $1$ | $9$ | $F_{5}\times C_2$ | $[\ ]_{10}^{4}$ | |
| $5$ | 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 | $x^{6} - 10 x^{4} + 25 x^{2} - 500$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| $7$ | 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.5.4.1 | $x^{5} - 7$ | $5$ | $1$ | $4$ | $F_5$ | $[\ ]_{5}^{4}$ | |
| 7.6.0.1 | $x^{6} + 3 x^{2} - x + 5$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| $11$ | 11.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 11.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.3.2.1 | $x^{3} - 11$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 11.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 11.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 107 | Data not computed | ||||||
| 239 | Data not computed | ||||||
| 1979 | Data not computed | ||||||
| 112901 | Data not computed | ||||||