Normalized defining polynomial
\( x^{35} + 4 x - 4 \)
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: | \(-2194316076337141133070751805670925124005717337380318127214034944=-\,2^{34}\cdot 3\cdot 227\cdot 499\cdot 1637\cdot 252983\cdot 13122193474139\cdot 69164815108090306157070281\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $64.53$ | 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: | $2, 3, 227, 499, 1637, 252983, 13122193474139, 69164815108090306157070281$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2} a^{18}$, $\frac{1}{2} a^{19}$, $\frac{1}{2} a^{20}$, $\frac{1}{2} a^{21}$, $\frac{1}{2} a^{22}$, $\frac{1}{2} a^{23}$, $\frac{1}{2} a^{24}$, $\frac{1}{2} a^{25}$, $\frac{1}{2} a^{26}$, $\frac{1}{2} a^{27}$, $\frac{1}{2} a^{28}$, $\frac{1}{2} a^{29}$, $\frac{1}{2} a^{30}$, $\frac{1}{2} a^{31}$, $\frac{1}{2} a^{32}$, $\frac{1}{2} a^{33}$, $\frac{1}{6} a^{34} - \frac{1}{6} a^{33} + \frac{1}{6} a^{32} - \frac{1}{6} a^{31} + \frac{1}{6} a^{30} - \frac{1}{6} a^{29} + \frac{1}{6} a^{28} - \frac{1}{6} a^{27} + \frac{1}{6} a^{26} - \frac{1}{6} a^{25} + \frac{1}{6} a^{24} - \frac{1}{6} a^{23} + \frac{1}{6} a^{22} - \frac{1}{6} a^{21} + \frac{1}{6} a^{20} - \frac{1}{6} a^{19} + \frac{1}{6} a^{18} + \frac{1}{3} a^{17} - \frac{1}{3} a^{16} + \frac{1}{3} a^{15} - \frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$
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 | R | R | $31{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ | $18{,}\,{\href{/LocalNumberField/7.14.0.1}{14} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ | $22{,}\,{\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ | $28{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ | $17{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ | $22{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ | $34{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ | $22{,}\,{\href{/LocalNumberField/29.11.0.1}{11} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ | $19{,}\,{\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ | $19{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ | $21{,}\,{\href{/LocalNumberField/41.11.0.1}{11} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ | $19{,}\,{\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ | $23{,}\,{\href{/LocalNumberField/47.11.0.1}{11} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ | $19{,}\,16$ | $27{,}\,{\href{/LocalNumberField/59.8.0.1}{8} }$ |
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 | $x^{12} - x^{4} - x^{3} - x^{2} + x - 1$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
| 227 | Data not computed | ||||||
| 499 | Data not computed | ||||||
| 1637 | Data not computed | ||||||
| 252983 | Data not computed | ||||||
| 13122193474139 | Data not computed | ||||||
| 69164815108090306157070281 | Data not computed | ||||||