Normalized defining polynomial
\( x^{33} + 5 x - 4 \)
Invariants
| Degree: | $33$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
| Signature: | $[1, 16]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
| |
| Discriminant: | \(102\!\cdots\!432\)\(\medspace = 2^{64}\cdot 163\cdot 181\cdot 50551\cdot 28444257707\cdot 1304516704756865886343705176787\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | $123.36$ | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
| |
| Ramified primes: | $2, 163, 181, 50551, 28444257707, 1304516704756865886343705176787$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
| |
| $|\Aut(K/\Q)|$: | $1$ | ||
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $\frac{1}{13} a^{32} - \frac{6}{13} a^{31} - \frac{3}{13} a^{30} + \frac{5}{13} a^{29} - \frac{4}{13} a^{28} - \frac{2}{13} a^{27} - \frac{1}{13} a^{26} + \frac{6}{13} a^{25} + \frac{3}{13} a^{24} - \frac{5}{13} a^{23} + \frac{4}{13} a^{22} + \frac{2}{13} a^{21} + \frac{1}{13} a^{20} - \frac{6}{13} a^{19} - \frac{3}{13} a^{18} + \frac{5}{13} a^{17} - \frac{4}{13} a^{16} - \frac{2}{13} a^{15} - \frac{1}{13} a^{14} + \frac{6}{13} a^{13} + \frac{3}{13} a^{12} - \frac{5}{13} a^{11} + \frac{4}{13} a^{10} + \frac{2}{13} a^{9} + \frac{1}{13} a^{8} - \frac{6}{13} a^{7} - \frac{3}{13} a^{6} + \frac{5}{13} a^{5} - \frac{4}{13} a^{4} - \frac{2}{13} a^{3} - \frac{1}{13} a^{2} + \frac{6}{13} a - \frac{5}{13}$
Class group and class number
not computed
Unit group
| Rank: | $16$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
| |
| Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
| |
| Fundamental units: | not computed | sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
| |
| Regulator: | not computed | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
Galois group
$S_{33}$ (as 33T162):
| A non-solvable group of order 8683317618811886495518194401280000000 |
| The 10143 conjugacy class representatives for $S_{33}$ are not computed |
| Character table for $S_{33}$ 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 | $26{,}\,{\href{/LocalNumberField/3.7.0.1}{7} }$ | ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ | $32{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ | $26{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{3}$ | $17{,}\,{\href{/LocalNumberField/13.13.0.1}{13} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ | $30{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ | $22{,}\,{\href{/LocalNumberField/19.7.0.1}{7} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }$ | $30{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ | $17{,}\,{\href{/LocalNumberField/29.11.0.1}{11} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ | $15{,}\,{\href{/LocalNumberField/31.9.0.1}{9} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ | $29{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ | $27{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ | $16{,}\,{\href{/LocalNumberField/43.13.0.1}{13} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ | $22{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ | ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.9.0.1}{9} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ | ${\href{/LocalNumberField/59.14.0.1}{14} }{,}\,{\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
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 | Data not computed | ||||||
| 163 | Data not computed | ||||||
| $181$ | $\Q_{181}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{181}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 181.2.1.1 | $x^{2} - 181$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 181.7.0.1 | $x^{7} - x + 2$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
| 181.8.0.1 | $x^{8} - x + 23$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 181.14.0.1 | $x^{14} - x + 2$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | |
| 50551 | Data not computed | ||||||
| 28444257707 | Data not computed | ||||||
| 1304516704756865886343705176787 | Data not computed | ||||||