Normalized defining polynomial
\( x^{32} - 2 x + 5 \)
Invariants
| Degree: | $32$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 16]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(1701411834604673989430707740777023676299648164966369654411826462457856=2^{30}\cdot 149\cdot 683\cdot 10501553\cdot 850518758389\cdot 1743272110648640538713595392906784971\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $145.70$ | 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: | $2, 149, 683, 10501553, 850518758389, 1743272110648640538713595392906784971$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $\frac{1}{2} a^{31} - \frac{1}{2} a^{30} - \frac{1}{2} a^{29} - \frac{1}{2} a^{28} - \frac{1}{2} a^{27} - \frac{1}{2} a^{26} - \frac{1}{2} a^{25} - \frac{1}{2} a^{24} - \frac{1}{2} a^{23} - \frac{1}{2} a^{22} - \frac{1}{2} a^{21} - \frac{1}{2} a^{20} - \frac{1}{2} a^{19} - \frac{1}{2} a^{18} - \frac{1}{2} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$
Class group and class number
Not computed
Unit group
| Rank: | $15$ | 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: | Not computed | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | Not computed | 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 | $28{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ | ${\href{/LocalNumberField/5.3.0.1}{3} }^{10}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ | $19{,}\,{\href{/LocalNumberField/7.7.0.1}{7} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }$ | $24{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ | $20{,}\,{\href{/LocalNumberField/13.7.0.1}{7} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ | $18{,}\,{\href{/LocalNumberField/17.7.0.1}{7} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ | $32$ | ${\href{/LocalNumberField/23.13.0.1}{13} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ | $27{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ | $16^{2}$ | $24{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/LocalNumberField/41.11.0.1}{11} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ | $15{,}\,{\href{/LocalNumberField/43.11.0.1}{11} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ | $29{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ | $32$ | $21{,}\,{\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$ |
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 | ||||||
| $149$ | 149.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 149.2.1.1 | $x^{2} - 149$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 149.3.0.1 | $x^{3} - x + 18$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 149.10.0.1 | $x^{10} - x + 71$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
| 149.15.0.1 | $x^{15} - x + 11$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | |
| 683 | Data not computed | ||||||
| 10501553 | Data not computed | ||||||
| 850518758389 | Data not computed | ||||||
| 1743272110648640538713595392906784971 | Data not computed | ||||||