Normalized defining polynomial
\( x^{35} + 4 x - 2 \)
Invariants
| Degree: | $35$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
| Signature: | $[1, 17]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
| |
| Discriminant: | \(-16\!\cdots\!904\)\(\medspace = -\,2^{34}\cdot 3\cdot 291647\cdot 188120963917\cdot 64268062741853\cdot 129806582432617\cdot 699615044068223\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | $101.44$ | 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: | $2, 3, 291647, 188120963917, 64268062741853, 129806582432617, 699615044068223$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $\frac{1}{29} a^{34} - \frac{11}{29} a^{33} + \frac{5}{29} a^{32} + \frac{3}{29} a^{31} - \frac{4}{29} a^{30} - \frac{14}{29} a^{29} + \frac{9}{29} a^{28} - \frac{12}{29} a^{27} - \frac{13}{29} a^{26} - \frac{2}{29} a^{25} - \frac{7}{29} a^{24} - \frac{10}{29} a^{23} - \frac{6}{29} a^{22} + \frac{8}{29} a^{21} - \frac{1}{29} a^{20} + \frac{11}{29} a^{19} - \frac{5}{29} a^{18} - \frac{3}{29} a^{17} + \frac{4}{29} a^{16} + \frac{14}{29} a^{15} - \frac{9}{29} a^{14} + \frac{12}{29} a^{13} + \frac{13}{29} a^{12} + \frac{2}{29} a^{11} + \frac{7}{29} a^{10} + \frac{10}{29} a^{9} + \frac{6}{29} a^{8} - \frac{8}{29} a^{7} + \frac{1}{29} a^{6} - \frac{11}{29} a^{5} + \frac{5}{29} a^{4} + \frac{3}{29} a^{3} - \frac{4}{29} a^{2} - \frac{14}{29} a + \frac{13}{29}$
Class group and class number
not computed
Unit group
| Rank: | $17$ | 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
$S_{35}$ (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 | R | R | $26{,}\,{\href{/LocalNumberField/5.7.0.1}{7} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ | $21{,}\,{\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ | $32{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ | ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.7.0.1}{7} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/17.14.0.1}{14} }{,}\,{\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ | $19{,}\,{\href{/LocalNumberField/19.11.0.1}{11} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | $29{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ | $17{,}\,16{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | $21{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | $15{,}\,{\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ | $27{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ | $34{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ | ${\href{/LocalNumberField/47.14.0.1}{14} }{,}\,{\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/53.11.0.1}{11} }{,}\,{\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.7.0.1}{7} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ | $23{,}\,{\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$ |
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 | ||||||
| $3$ | 3.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.8.0.1 | $x^{8} - x^{3} + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 3.11.0.1 | $x^{11} + x^{2} - x + 1$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | |
| 3.12.0.1 | $x^{12} - x^{4} - x^{3} - x^{2} + x - 1$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
| 291647 | Data not computed | ||||||
| 188120963917 | Data not computed | ||||||
| 64268062741853 | Data not computed | ||||||
| 129806582432617 | Data not computed | ||||||
| 699615044068223 | Data not computed | ||||||