Normalized defining polynomial
\( x^{33} - 3 x - 4 \)
Invariants
| Degree: | $33$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[1, 16]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(48605137597330374187208995479905976865346972484441609677173099069440=2^{64}\cdot 3^{33}\cdot 5\cdot 7\cdot 211\cdot 47701\cdot 56203340663\cdot 23939822997841\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $112.49$ | 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: | $2, 3, 5, 7, 211, 47701, 56203340663, 23939822997841$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $\frac{1}{7} a^{32} + \frac{3}{7} a^{31} + \frac{2}{7} a^{30} - \frac{1}{7} a^{29} - \frac{3}{7} a^{28} - \frac{2}{7} a^{27} + \frac{1}{7} a^{26} + \frac{3}{7} a^{25} + \frac{2}{7} a^{24} - \frac{1}{7} a^{23} - \frac{3}{7} a^{22} - \frac{2}{7} a^{21} + \frac{1}{7} a^{20} + \frac{3}{7} a^{19} + \frac{2}{7} a^{18} - \frac{1}{7} a^{17} - \frac{3}{7} a^{16} - \frac{2}{7} a^{15} + \frac{1}{7} a^{14} + \frac{3}{7} a^{13} + \frac{2}{7} a^{12} - \frac{1}{7} a^{11} - \frac{3}{7} a^{10} - \frac{2}{7} a^{9} + \frac{1}{7} a^{8} + \frac{3}{7} a^{7} + \frac{2}{7} a^{6} - \frac{1}{7} a^{5} - \frac{3}{7} a^{4} - \frac{2}{7} a^{3} + \frac{1}{7} a^{2} + \frac{3}{7} a - \frac{1}{7}$
Class group and class number
Not computed
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: | 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
$S_{33}$ (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 | R | R | R | ${\href{/LocalNumberField/11.13.0.1}{13} }{,}\,{\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ | $29{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ | $30{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ | $22{,}\,{\href{/LocalNumberField/19.7.0.1}{7} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }$ | $30{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ | $21{,}\,{\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ | $24{,}\,{\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ | ${\href{/LocalNumberField/37.11.0.1}{11} }{,}\,{\href{/LocalNumberField/37.9.0.1}{9} }{,}\,{\href{/LocalNumberField/37.7.0.1}{7} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ | $26{,}\,{\href{/LocalNumberField/41.7.0.1}{7} }$ | $30{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ | $22{,}\,{\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ | $28{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ | $17{,}\,16$ |
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 | ||||||
| $3$ | 3.3.3.1 | $x^{3} + 6 x + 3$ | $3$ | $1$ | $3$ | $S_3$ | $[3/2]_{2}$ |
| 3.15.15.39 | $x^{15} + 12 x^{14} + 15 x^{13} + 9 x^{11} + 12 x^{10} + 10 x^{9} + 6 x^{8} + 9 x^{7} + 18 x^{6} + 9 x^{5} + 12 x^{4} + 22 x^{3} + 12 x^{2} + 18 x + 1$ | $3$ | $5$ | $15$ | 15T44 | $[3/2, 3/2, 3/2, 3/2, 3/2]_{2}^{5}$ | |
| 3.15.15.2 | $x^{15} + 3 x^{14} + 15 x^{13} + 24 x^{12} + 24 x^{11} + 18 x^{10} + 10 x^{9} + 21 x^{8} + 9 x^{7} + 15 x^{6} + 9 x^{5} + 3 x^{4} + 13 x^{3} + 18 x + 7$ | $3$ | $5$ | $15$ | 15T44 | $[3/2, 3/2, 3/2, 3/2, 3/2]_{2}^{5}$ | |
| 5 | Data not computed | ||||||
| $7$ | 7.2.1.1 | $x^{2} - 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 7.6.0.1 | $x^{6} + 3 x^{2} - x + 5$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 7.6.0.1 | $x^{6} + 3 x^{2} - x + 5$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 7.7.0.1 | $x^{7} - x + 2$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
| 7.12.0.1 | $x^{12} + 3 x^{2} - 2 x + 3$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
| 211 | Data not computed | ||||||
| 47701 | Data not computed | ||||||
| 56203340663 | Data not computed | ||||||
| 23939822997841 | Data not computed | ||||||