Normalized defining polynomial
\( x^{32} - x - 3 \)
Invariants
| Degree: | $32$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[2, 15]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-488226435913860575293430098418591721500242319636808923916247=-\,17\cdot 18793\cdot 441552440027\cdot 34029561295774139\cdot 101703902339059152088627079\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $73.33$ | 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: | $17, 18793, 441552440027, 34029561295774139, 101703902339059152088627079$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $\frac{1}{43} a^{31} + \frac{8}{43} a^{30} + \frac{21}{43} a^{29} - \frac{4}{43} a^{28} + \frac{11}{43} a^{27} + \frac{2}{43} a^{26} + \frac{16}{43} a^{25} - \frac{1}{43} a^{24} - \frac{8}{43} a^{23} - \frac{21}{43} a^{22} + \frac{4}{43} a^{21} - \frac{11}{43} a^{20} - \frac{2}{43} a^{19} - \frac{16}{43} a^{18} + \frac{1}{43} a^{17} + \frac{8}{43} a^{16} + \frac{21}{43} a^{15} - \frac{4}{43} a^{14} + \frac{11}{43} a^{13} + \frac{2}{43} a^{12} + \frac{16}{43} a^{11} - \frac{1}{43} a^{10} - \frac{8}{43} a^{9} - \frac{21}{43} a^{8} + \frac{4}{43} a^{7} - \frac{11}{43} a^{6} - \frac{2}{43} a^{5} - \frac{16}{43} a^{4} + \frac{1}{43} a^{3} + \frac{8}{43} a^{2} + \frac{21}{43} a - \frac{5}{43}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $16$ | 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: | \( 228566719485828220 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$S_{32}$ (as 32T2801324):
| A non-solvable group of order 263130836933693530167218012160000000 |
| The 8349 conjugacy class representatives for $S_{32}$ are not computed |
| Character table for $S_{32}$ is not computed |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.10.0.1}{10} }^{3}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ | $30{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ | $26{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ | $32$ | $17{,}\,{\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ | $18{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ | R | $24{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ | $31{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ | $20{,}\,{\href{/LocalNumberField/29.12.0.1}{12} }$ | ${\href{/LocalNumberField/31.8.0.1}{8} }^{4}$ | ${\href{/LocalNumberField/37.14.0.1}{14} }{,}\,{\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ | ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ | $23{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ | $27{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }$ | $24{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ | $15{,}\,{\href{/LocalNumberField/59.13.0.1}{13} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }$ |
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 |
|---|---|---|---|---|---|---|---|
| 17 | Data not computed | ||||||
| 18793 | Data not computed | ||||||
| 441552440027 | Data not computed | ||||||
| 34029561295774139 | Data not computed | ||||||
| 101703902339059152088627079 | Data not computed | ||||||