Normalized defining polynomial
\( x^{15} - 145860 x^{10} - 243079056 x^{5} - 4696791102432 \)
Invariants
| Degree: | $15$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[1, 7]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-5617330134094061678662632074276610010758000000000=-\,2^{10}\cdot 3^{12}\cdot 5^{9}\cdot 11^{13}\cdot 10889^{5}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $1778.15$ | 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: | $2, 3, 5, 11, 10889$ | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{6} a^{5}$, $\frac{1}{12} a^{6}$, $\frac{1}{12} a^{7}$, $\frac{1}{12} a^{8}$, $\frac{1}{24} a^{9}$, $\frac{1}{59263665912} a^{10} + \frac{12556691}{448967166} a^{5} + \frac{5679428}{74827861}$, $\frac{1}{770427656856} a^{11} - \frac{2395045}{224483583} a^{6} + \frac{454646594}{972762193} a$, $\frac{1}{1101711549304080} a^{12} + \frac{1}{148159164780} a^{10} - \frac{1}{60} a^{9} + \frac{1}{30} a^{8} - \frac{17290025981}{642023047380} a^{7} - \frac{16571493}{748278610} a^{5} + \frac{1}{10} a^{4} + \frac{1}{5} a^{3} + \frac{163651371721}{695524967995} a^{2} + \frac{2}{5} a + \frac{11358856}{374139305}$, $\frac{1}{14322250140953040} a^{13} + \frac{1}{1926069142140} a^{11} - \frac{1}{148159164780} a^{10} - \frac{1}{60} a^{9} + \frac{303721497709}{8346299615940} a^{8} - \frac{31329407}{1496557220} a^{6} - \frac{29128184}{374139305} a^{5} - \frac{1}{10} a^{4} - \frac{3150322096533}{18083649167870} a^{3} - \frac{1}{5} a^{2} + \frac{1882055381}{4863810965} a + \frac{12693801}{74827861}$, $\frac{1}{2048081770156284720} a^{14} - \frac{1}{1926069142140} a^{11} - \frac{6373318195043}{1193520845079420} a^{9} + \frac{1}{30} a^{8} + \frac{1}{60} a^{7} - \frac{55667501}{4489671660} a^{6} - \frac{61017999433717}{2585961831005410} a^{4} - \frac{1}{5} a^{3} - \frac{2}{5} a^{2} + \frac{2008993391}{4863810965} a + \frac{1}{5}$
Class group and class number
$C_{165}$, which has order $165$ (assuming GRH)
Unit group
| Rank: | $7$ | 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: | \( 49247044495643600 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$S_3\times F_5$ (as 15T11):
| A solvable group of order 120 |
| The 15 conjugacy class representatives for $F_5 \times S_3$ |
| Character table for $F_5 \times S_3$ |
Intermediate fields
| 3.1.479116.1, 5.1.148240125.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 30 siblings: | 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 | ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ | R | ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{5}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ | $15$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$ |
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 | $x^{3} - 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 2.12.8.1 | $x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$ | $3$ | $4$ | $8$ | $C_3 : C_4$ | $[\ ]_{3}^{4}$ | |
| $3$ | 3.5.4.1 | $x^{5} - 3$ | $5$ | $1$ | $4$ | $F_5$ | $[\ ]_{5}^{4}$ |
| 3.10.8.1 | $x^{10} - 3 x^{5} + 18$ | $5$ | $2$ | $8$ | $F_5$ | $[\ ]_{5}^{4}$ | |
| $5$ | 5.3.0.1 | $x^{3} - x + 2$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 5.12.9.2 | $x^{12} - 10 x^{8} + 25 x^{4} - 500$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ | |
| $11$ | 11.5.4.5 | $x^{5} - 99$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |
| 11.10.9.4 | $x^{10} - 99$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ | |
| 10889 | Data not computed | ||||||