Normalized defining polynomial
\( x^{15} - 81 x^{13} + 1163 x^{11} - 608 x^{10} - 6705 x^{9} + 23976 x^{8} + 253668 x^{7} - 1670632 x^{6} + 1782240 x^{5} + 3051360 x^{4} - 4986800 x^{3} - 894400 x^{2} + 2912000 x - 832000 \)
Invariants
| Degree: | $15$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[3, 6]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(12508510250549402266838396233711616000000=2^{23}\cdot 5^{6}\cdot 13^{4}\cdot 37^{5}\cdot 151^{2}\cdot 457^{2}\cdot 3181^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $471.14$ | 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: | $2, 5, 13, 37, 151, 457, 3181$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{2} a^{7} + \frac{3}{8} a^{6} - \frac{1}{2} a^{5} - \frac{1}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{40} a^{11} - \frac{1}{40} a^{9} + \frac{3}{40} a^{7} - \frac{1}{5} a^{6} + \frac{3}{8} a^{5} + \frac{2}{5} a^{4} - \frac{3}{10} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{80} a^{12} - \frac{1}{80} a^{10} + \frac{3}{80} a^{8} + \frac{2}{5} a^{7} - \frac{5}{16} a^{6} - \frac{3}{10} a^{5} + \frac{7}{20} a^{4} + \frac{1}{10} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{800} a^{13} - \frac{1}{800} a^{11} + \frac{83}{800} a^{9} + \frac{6}{25} a^{8} + \frac{27}{160} a^{7} + \frac{17}{100} a^{6} - \frac{33}{200} a^{5} + \frac{31}{100} a^{4} - \frac{3}{20} a^{3} - \frac{1}{2} a$, $\frac{1}{2933440045792386799062320819019593600} a^{14} + \frac{16720951773777894457292638088321}{366680005724048349882790102377449200} a^{13} - \frac{18207692270115645385343818047599201}{2933440045792386799062320819019593600} a^{12} + \frac{1386016089854826907924264357073389}{366680005724048349882790102377449200} a^{11} - \frac{139551114788986653298283634314206117}{2933440045792386799062320819019593600} a^{10} - \frac{6137890852140510313433894060711143}{366680005724048349882790102377449200} a^{9} - \frac{436090787164287413301871908243277409}{2933440045792386799062320819019593600} a^{8} + \frac{68944187066449068793756403119885691}{183340002862024174941395051188724600} a^{7} + \frac{190212924944852908744614476771161069}{733360011448096699765580204754898400} a^{6} + \frac{100273231932653740034233872005115409}{366680005724048349882790102377449200} a^{5} - \frac{41688407723353190173580301375813993}{91670001431012087470697525594362300} a^{4} + \frac{7442956828133985566377746034012063}{18334000286202417494139505118872460} a^{3} + \frac{5741059430273749983332162962694453}{36668000572404834988279010237744920} a^{2} - \frac{198142128718645846582205957549883}{1833400028620241749413950511887246} a + \frac{177074536003452509751009105652809}{916700014310120874706975255943623}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $8$ | 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: | \( 1959807158240000 \) (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
| 3.3.148.1 |
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 | ${\href{/LocalNumberField/3.9.0.1}{9} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }$ | R | ${\href{/LocalNumberField/7.9.0.1}{9} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ | R | ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ | ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ | ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ | R | ${\href{/LocalNumberField/41.9.0.1}{9} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ | ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| $2$ | 2.3.2.1 | $x^{3} - 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 2.6.10.4 | $x^{6} + 2 x^{5} + 2 x^{4} + 6$ | $6$ | $1$ | $10$ | $S_4$ | $[8/3, 8/3]_{3}^{2}$ | |
| 2.6.11.8 | $x^{6} + 4 x^{4} + 2 x^{2} + 6$ | $6$ | $1$ | $11$ | $S_4\times C_2$ | $[8/3, 8/3, 3]_{3}^{2}$ | |
| $5$ | $\Q_{5}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $13$ | 13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.5.4.1 | $x^{5} - 13$ | $5$ | $1$ | $4$ | $F_5$ | $[\ ]_{5}^{4}$ | |
| 13.6.0.1 | $x^{6} + x^{2} - 2 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 37 | Data not computed | ||||||
| 151 | Data not computed | ||||||
| 457 | Data not computed | ||||||
| 3181 | Data not computed | ||||||