Normalized defining polynomial
\( x^{46} - 2 x + 5 \)
Invariants
| Degree: | $46$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 23]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-217996971946336332095812748918644261487886130507332474040110020000000000000000000000000000000000000000000000=-\,2^{47}\cdot 5^{46}\cdot 11\cdot 839\cdot 8329\cdot 63643414721\cdot 3939071764787569\cdot 565621085623946238157010489\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $215.50$ | 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: | $2, 5, 11, 839, 8329, 63643414721, 3939071764787569, 565621085623946238157010489$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $a^{42}$, $a^{43}$, $a^{44}$, $\frac{1}{2} a^{45} - \frac{1}{2} a^{44} - \frac{1}{2} a^{43} - \frac{1}{2} a^{42} - \frac{1}{2} a^{41} - \frac{1}{2} a^{40} - \frac{1}{2} a^{39} - \frac{1}{2} a^{38} - \frac{1}{2} a^{37} - \frac{1}{2} a^{36} - \frac{1}{2} a^{35} - \frac{1}{2} a^{34} - \frac{1}{2} a^{33} - \frac{1}{2} a^{32} - \frac{1}{2} a^{31} - \frac{1}{2} a^{30} - \frac{1}{2} a^{29} - \frac{1}{2} a^{28} - \frac{1}{2} a^{27} - \frac{1}{2} a^{26} - \frac{1}{2} a^{25} - \frac{1}{2} a^{24} - \frac{1}{2} a^{23} - \frac{1}{2} a^{22} - \frac{1}{2} a^{21} - \frac{1}{2} a^{20} - \frac{1}{2} a^{19} - \frac{1}{2} a^{18} - \frac{1}{2} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$
Class group and class number
Not computed
Unit group
| Rank: | $22$ | 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
$S_{46}$ (as 46T56):
| A non-solvable group of order 5502622159812088949850305428800254892961651752960000000000 |
| The 105558 conjugacy class representatives for $S_{46}$ are not computed |
| Character table for $S_{46}$ 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 | $36{,}\,{\href{/LocalNumberField/3.10.0.1}{10} }$ | R | $32{,}\,{\href{/LocalNumberField/7.7.0.1}{7} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ | R | $34{,}\,{\href{/LocalNumberField/13.12.0.1}{12} }$ | $16{,}\,{\href{/LocalNumberField/17.14.0.1}{14} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ | $24{,}\,{\href{/LocalNumberField/19.14.0.1}{14} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | $27{,}\,{\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ | $44{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ | $29{,}\,{\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | $39{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ | $24{,}\,{\href{/LocalNumberField/41.11.0.1}{11} }{,}\,{\href{/LocalNumberField/41.9.0.1}{9} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ | ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.11.0.1}{11} }{,}\,{\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ | $23{,}\,{\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ | $17{,}\,{\href{/LocalNumberField/53.12.0.1}{12} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ | $41{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$ |
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 | ||||||
| 5 | Data not computed | ||||||
| 11 | Data not computed | ||||||
| 839 | Data not computed | ||||||
| 8329 | Data not computed | ||||||
| 63643414721 | Data not computed | ||||||
| 3939071764787569 | Data not computed | ||||||
| 565621085623946238157010489 | Data not computed | ||||||