Normalized defining polynomial
\( x^{15} + 160 x^{11} - 128 x^{10} + 5200 x^{7} - 8320 x^{6} + 3328 x^{5} - 14500 x^{3} + 34800 x^{2} - 27840 x + 7424 \)
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: | \(3138922448058688000000000000000000=2^{24}\cdot 5^{18}\cdot 29^{4}\cdot 37^{5}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $171.05$ | 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: | $2, 5, 29, 37$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} - \frac{1}{5}$, $\frac{1}{10} a^{6} + \frac{2}{5} a$, $\frac{1}{20} a^{7} + \frac{1}{5} a^{2} - \frac{1}{2} a$, $\frac{1}{20} a^{8} + \frac{1}{5} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{20} a^{9} + \frac{1}{5} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{100} a^{10} + \frac{2}{25} a^{5} - \frac{1}{2} a^{4} + \frac{4}{25}$, $\frac{1}{100} a^{11} - \frac{1}{50} a^{6} - \frac{1}{10} a^{5} - \frac{6}{25} a - \frac{2}{5}$, $\frac{1}{100} a^{12} - \frac{1}{50} a^{7} - \frac{6}{25} a^{2}$, $\frac{1}{51200} a^{13} + \frac{13}{6400} a^{12} - \frac{1}{640} a^{11} - \frac{1}{200} a^{10} - \frac{7}{320} a^{9} + \frac{1}{400} a^{8} + \frac{1}{100} a^{7} + \frac{197}{3200} a^{5} + \frac{1}{10} a^{4} - \frac{21}{50} a^{3} + \frac{8}{25} a^{2} + \frac{299}{2560} a + \frac{469}{3200}$, $\frac{1}{26214400} a^{14} - \frac{51}{6553600} a^{13} + \frac{2601}{1638400} a^{12} - \frac{1579}{409600} a^{11} - \frac{2931}{819200} a^{10} - \frac{3}{25600} a^{9} + \frac{153}{6400} a^{8} + \frac{37}{1600} a^{7} + \frac{69957}{1638400} a^{6} - \frac{12609}{409600} a^{5} + \frac{17}{50} a^{4} + \frac{7}{50} a^{3} + \frac{258519}{6553600} a^{2} - \frac{103083}{819200} a + \frac{11887}{409600}$
Class group and class number
$C_{3}$, which has order $3$ (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: | \( 70532799768.3 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 1296000 |
| The 65 conjugacy class representatives for [A(5)^3]S(3)=A(5)wrS(3) are not computed |
| Character table for [A(5)^3]S(3)=A(5)wrS(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 | ${\href{/LocalNumberField/3.9.0.1}{9} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ | R | ${\href{/LocalNumberField/7.9.0.1}{9} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ | $15$ | ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ | ${\href{/LocalNumberField/19.5.0.1}{5} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ | ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ | R | ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ | R | ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ | ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }$ | $15$ | ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ | ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$ |
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 | $x^{3} - 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 2.12.22.115 | $x^{12} + 2 x^{11} - 2 x^{10} - 2 x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 2 x^{4} - 2 x^{2} - 2$ | $12$ | $1$ | $22$ | 12T68 | $[4/3, 4/3, 8/3, 8/3]_{3}^{2}$ | |
| $5$ | 5.5.6.2 | $x^{5} + 15 x^{2} + 5$ | $5$ | $1$ | $6$ | $D_{5}$ | $[3/2]_{2}$ |
| 5.10.12.8 | $x^{10} + 10 x^{8} + 15 x^{7} + 15 x^{6} - 20 x^{5} + 5 x^{4} + 5 x^{2} - 10 x + 7$ | $5$ | $2$ | $12$ | $F_5$ | $[3/2]_{2}^{2}$ | |
| $29$ | 29.5.4.1 | $x^{5} - 29$ | $5$ | $1$ | $4$ | $D_{5}$ | $[\ ]_{5}^{2}$ |
| 29.10.0.1 | $x^{10} + x^{2} - 2 x + 2$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
| 37 | Data not computed | ||||||