Normalized defining polynomial
\( x^{35} + 5 x - 3 \)
Invariants
| Degree: | $35$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[1, 17]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-1368653637240265747064304367405255252659852182698564487509429454803466796875=-\,5^{33}\cdot 1098074000209\cdot 10706605584650121566295636839261553663967\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $140.20$ | 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: | $5, 1098074000209, 10706605584650121566295636839261553663967$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $\frac{1}{5} a^{34} + \frac{2}{5} a^{33} - \frac{1}{5} a^{32} - \frac{2}{5} a^{31} + \frac{1}{5} a^{30} + \frac{2}{5} a^{29} - \frac{1}{5} a^{28} - \frac{2}{5} a^{27} + \frac{1}{5} a^{26} + \frac{2}{5} a^{25} - \frac{1}{5} a^{24} - \frac{2}{5} a^{23} + \frac{1}{5} a^{22} + \frac{2}{5} a^{21} - \frac{1}{5} a^{20} - \frac{2}{5} a^{19} + \frac{1}{5} a^{18} + \frac{2}{5} a^{17} - \frac{1}{5} a^{16} - \frac{2}{5} a^{15} + \frac{1}{5} a^{14} + \frac{2}{5} a^{13} - \frac{1}{5} a^{12} - \frac{2}{5} a^{11} + \frac{1}{5} a^{10} + \frac{2}{5} a^{9} - \frac{1}{5} a^{8} - \frac{2}{5} a^{7} + \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a - \frac{1}{5}$
Class group and class number
Not computed
Unit group
| Rank: | $17$ | 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
$S_{35}$ (as 35T407):
| A non-solvable group of order 10333147966386144929666651337523200000000 |
| The 14883 conjugacy class representatives for $S_{35}$ are not computed |
| Character table for $S_{35}$ 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 | $33{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ | $16^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{3}$ | R | $26{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ | $28{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ | $17{,}\,16{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ | $19{,}\,15{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ | $18{,}\,15{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ | $26{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ | $32{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ | $20{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ | $26{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ | $19{,}\,{\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }$ | $15{,}\,{\href{/LocalNumberField/43.13.0.1}{13} }{,}\,{\href{/LocalNumberField/43.7.0.1}{7} }$ | $27{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ | $21{,}\,{\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/59.13.0.1}{13} }{,}\,{\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.7.0.1}{7} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$ |
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 | ||||||
| 1098074000209 | Data not computed | ||||||
| 10706605584650121566295636839261553663967 | Data not computed | ||||||