Normalized defining polynomial
\( x^{41} + 5 x - 2 \)
Invariants
| Degree: | $41$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[1, 20]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(137438953472000365828857506909812624694004283442925067740090273256794495510342500893383983104=2^{38}\cdot 163\cdot 743\cdot 10289\cdot 104597\cdot 31257021937\cdot 13675604446553671\cdot 95253739993542479\cdot 94216078858946850824041\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $176.71$ | 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: | $2, 163, 743, 10289, 104597, 31257021937, 13675604446553671, 95253739993542479, 94216078858946850824041$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $\frac{1}{2} a^{40} - \frac{1}{2} a^{39} - \frac{1}{2} a^{38} - \frac{1}{2} a^{37} - \frac{1}{2} a^{36} - \frac{1}{2} a^{35} - \frac{1}{2} a^{34} - \frac{1}{2} a^{33} - \frac{1}{2} a^{32} - \frac{1}{2} a^{31} - \frac{1}{2} a^{30} - \frac{1}{2} a^{29} - \frac{1}{2} a^{28} - \frac{1}{2} a^{27} - \frac{1}{2} a^{26} - \frac{1}{2} a^{25} - \frac{1}{2} a^{24} - \frac{1}{2} a^{23} - \frac{1}{2} a^{22} - \frac{1}{2} a^{21} - \frac{1}{2} a^{20} - \frac{1}{2} a^{19} - \frac{1}{2} a^{18} - \frac{1}{2} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$
Class group and class number
Not computed
Unit group
| Rank: | $20$ | 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
$S_{41}$ (as 41T10):
| A non-solvable group of order 33452526613163807108170062053440751665152000000000 |
| The 44583 conjugacy class representatives for $S_{41}$ are not computed |
| Character table for $S_{41}$ 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 | $41$ | $20^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ | $28{,}\,{\href{/LocalNumberField/7.7.0.1}{7} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }$ | $40{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ | $37{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ | $25{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ | $22{,}\,{\href{/LocalNumberField/19.13.0.1}{13} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }$ | $27{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ | $25{,}\,{\href{/LocalNumberField/29.14.0.1}{14} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | $27{,}\,{\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ | $22{,}\,{\href{/LocalNumberField/37.7.0.1}{7} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ | $20^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ | $26{,}\,{\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }$ | $26{,}\,{\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ | $38{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ | $25{,}\,{\href{/LocalNumberField/59.13.0.1}{13} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$ |
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 | ||||||
| 743 | Data not computed | ||||||
| 10289 | Data not computed | ||||||
| 104597 | Data not computed | ||||||
| 31257021937 | Data not computed | ||||||
| 13675604446553671 | Data not computed | ||||||
| 95253739993542479 | Data not computed | ||||||
| 94216078858946850824041 | Data not computed | ||||||