Normalized defining polynomial
\( x^{32} - 4 x - 4 \)
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: | \(-65703479140314228729138851088370269541878848552406351872=-\,2^{32}\cdot 7\cdot 683\cdot 64737929\cdot 49425487325967483301651503828482843\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $55.50$ | 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: | $2, 7, 683, 64737929, 49425487325967483301651503828482843$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16}$, $\frac{1}{2} a^{17}$, $\frac{1}{2} a^{18}$, $\frac{1}{2} a^{19}$, $\frac{1}{2} a^{20}$, $\frac{1}{2} a^{21}$, $\frac{1}{2} a^{22}$, $\frac{1}{2} a^{23}$, $\frac{1}{2} a^{24}$, $\frac{1}{2} a^{25}$, $\frac{1}{2} a^{26}$, $\frac{1}{2} a^{27}$, $\frac{1}{2} a^{28}$, $\frac{1}{2} a^{29}$, $\frac{1}{2} a^{30}$, $\frac{1}{10} a^{31} - \frac{1}{5} a^{30} - \frac{1}{10} a^{29} + \frac{1}{5} a^{28} + \frac{1}{10} a^{27} - \frac{1}{5} a^{26} - \frac{1}{10} a^{25} + \frac{1}{5} a^{24} + \frac{1}{10} a^{23} - \frac{1}{5} a^{22} - \frac{1}{10} a^{21} + \frac{1}{5} a^{20} + \frac{1}{10} a^{19} - \frac{1}{5} a^{18} - \frac{1}{10} a^{17} + \frac{1}{5} a^{16} - \frac{2}{5} a^{15} - \frac{1}{5} a^{14} + \frac{2}{5} a^{13} + \frac{1}{5} a^{12} - \frac{2}{5} a^{11} - \frac{1}{5} a^{10} + \frac{2}{5} a^{9} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2} + \frac{2}{5} a - \frac{1}{5}$
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: | \( 2729938722859418.5 \) (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 | R | $28{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ | $19{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ | R | $17{,}\,{\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ | $17{,}\,{\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ | $29{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ | $23{,}\,{\href{/LocalNumberField/19.9.0.1}{9} }$ | $30{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ | $19{,}\,{\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ | $30{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | $21{,}\,{\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ | $24{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ | $15{,}\,{\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ | ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}{,}\,{\href{/LocalNumberField/53.10.0.1}{10} }$ | $29{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$ |
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 | Data not computed | ||||||
| $7$ | $\Q_{7}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 7.2.1.2 | $x^{2} + 14$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.3.0.1 | $x^{3} - x + 2$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 7.3.0.1 | $x^{3} - x + 2$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 7.4.0.1 | $x^{4} + x^{2} - 3 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 7.4.0.1 | $x^{4} + x^{2} - 3 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 7.5.0.1 | $x^{5} - x + 4$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 7.10.0.1 | $x^{10} + 5 x^{2} - x + 5$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
| 683 | Data not computed | ||||||
| 64737929 | Data not computed | ||||||
| 49425487325967483301651503828482843 | Data not computed | ||||||