Normalized defining polynomial
\( x^{39} - x - 4 \)
Invariants
| Degree: | $39$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[1, 19]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-30949918466425163032455347262376548002357192825523150128829529791279710855=-\,5\cdot 7\cdot 41\cdot 659\cdot 2251\cdot 31793\cdot 2678539\cdot 118932731307271627\cdot 1435541730552723797020749364164204653\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $76.63$ | 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: | $5, 7, 41, 659, 2251, 31793, 2678539, 118932731307271627, 1435541730552723797020749364164204653$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{2} a^{20} - \frac{1}{2} a$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{23} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{24} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{25} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{26} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{27} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{28} - \frac{1}{2} a^{9}$, $\frac{1}{2} a^{29} - \frac{1}{2} a^{10}$, $\frac{1}{2} a^{30} - \frac{1}{2} a^{11}$, $\frac{1}{2} a^{31} - \frac{1}{2} a^{12}$, $\frac{1}{2} a^{32} - \frac{1}{2} a^{13}$, $\frac{1}{2} a^{33} - \frac{1}{2} a^{14}$, $\frac{1}{2} a^{34} - \frac{1}{2} a^{15}$, $\frac{1}{2} a^{35} - \frac{1}{2} a^{16}$, $\frac{1}{2} a^{36} - \frac{1}{2} a^{17}$, $\frac{1}{2} a^{37} - \frac{1}{2} a^{18}$, $\frac{1}{2} a^{38} - \frac{1}{2} a^{19}$
Class group and class number
Not computed
Unit group
| Rank: | $19$ | 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
$S_{39}$ (as 39T306):
| A non-solvable group of order 20397882081197443358640281739902897356800000000 |
| The 31185 conjugacy class representatives for $S_{39}$ are not computed |
| Character table for $S_{39}$ 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 | $36{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ | $27{,}\,{\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ | R | R | $34{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ | $39$ | $18{,}\,17{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ | $20{,}\,19$ | $22{,}\,{\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ | $22{,}\,17$ | $26{,}\,{\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ | $29{,}\,{\href{/LocalNumberField/37.7.0.1}{7} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ | R | $39$ | $15{,}\,{\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.7.0.1}{7} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ | $15{,}\,{\href{/LocalNumberField/53.13.0.1}{13} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ | $19^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
| 5 | Data not computed | ||||||
| 7 | Data not computed | ||||||
| 41 | Data not computed | ||||||
| 659 | Data not computed | ||||||
| 2251 | Data not computed | ||||||
| 31793 | Data not computed | ||||||
| 2678539 | Data not computed | ||||||
| 118932731307271627 | Data not computed | ||||||
| 1435541730552723797020749364164204653 | Data not computed | ||||||