Normalized defining polynomial
\( x^{46} - 2x + 3 \)
Invariants
Degree: | $46$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 23]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-749\!\cdots\!368\) \(\medspace = -\,2^{47}\cdot 3^{45}\cdot 40739891922760603\cdot 108253239392882437561\cdot 408511528412100333650099\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(121.40\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
Galois root discriminant: | not computed | ||
Ramified primes: | \(2\), \(3\), \(40739891922760603\), \(108253239392882437561\), \(408511528412100333650099\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-10809\!\cdots\!14102}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $a^{42}$, $a^{43}$, $a^{44}$, $\frac{1}{11}a^{45}+\frac{3}{11}a^{44}-\frac{2}{11}a^{43}+\frac{5}{11}a^{42}+\frac{4}{11}a^{41}+\frac{1}{11}a^{40}+\frac{3}{11}a^{39}-\frac{2}{11}a^{38}+\frac{5}{11}a^{37}+\frac{4}{11}a^{36}+\frac{1}{11}a^{35}+\frac{3}{11}a^{34}-\frac{2}{11}a^{33}+\frac{5}{11}a^{32}+\frac{4}{11}a^{31}+\frac{1}{11}a^{30}+\frac{3}{11}a^{29}-\frac{2}{11}a^{28}+\frac{5}{11}a^{27}+\frac{4}{11}a^{26}+\frac{1}{11}a^{25}+\frac{3}{11}a^{24}-\frac{2}{11}a^{23}+\frac{5}{11}a^{22}+\frac{4}{11}a^{21}+\frac{1}{11}a^{20}+\frac{3}{11}a^{19}-\frac{2}{11}a^{18}+\frac{5}{11}a^{17}+\frac{4}{11}a^{16}+\frac{1}{11}a^{15}+\frac{3}{11}a^{14}-\frac{2}{11}a^{13}+\frac{5}{11}a^{12}+\frac{4}{11}a^{11}+\frac{1}{11}a^{10}+\frac{3}{11}a^{9}-\frac{2}{11}a^{8}+\frac{5}{11}a^{7}+\frac{4}{11}a^{6}+\frac{1}{11}a^{5}+\frac{3}{11}a^{4}-\frac{2}{11}a^{3}+\frac{5}{11}a^{2}+\frac{4}{11}a-\frac{1}{11}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
not computed
Unit group
Rank: | $22$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | not computed | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | not computed | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $
Galois group
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 | R | $22{,}\,18{,}\,{\href{/padicField/5.6.0.1}{6} }$ | $32{,}\,{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ | $19{,}\,16{,}\,{\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | $25{,}\,{\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $26{,}\,16{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/19.13.0.1}{13} }{,}\,{\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | $41{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | $16{,}\,{\href{/padicField/29.10.0.1}{10} }^{2}{,}\,{\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | $36{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ | $44{,}\,{\href{/padicField/37.2.0.1}{2} }$ | $24{,}\,21{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $25{,}\,15{,}\,{\href{/padicField/43.6.0.1}{6} }$ | $19^{2}{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $31{,}\,{\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.14.0.1}{14} }{,}\,{\href{/padicField/59.13.0.1}{13} }{,}\,{\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.3.2 | $x^{2} + 4 x + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
Deg $22$ | $2$ | $11$ | $22$ | ||||
Deg $22$ | $2$ | $11$ | $22$ | ||||
\(3\) | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
3.9.9.11 | $x^{9} + 3 x + 3$ | $9$ | $1$ | $9$ | $(C_3^2:C_8):C_2$ | $[9/8, 9/8]_{8}^{2}$ | |
Deg $36$ | $9$ | $4$ | $36$ | ||||
\(40739891922760603\) | $\Q_{40739891922760603}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
Deg $21$ | $1$ | $21$ | $0$ | $C_{21}$ | $[\ ]^{21}$ | ||
\(108\!\cdots\!561\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $30$ | $1$ | $30$ | $0$ | $C_{30}$ | $[\ ]^{30}$ | ||
\(408\!\cdots\!099\) | $\Q_{40\!\cdots\!99}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $32$ | $1$ | $32$ | $0$ | 32T33 | $[\ ]^{32}$ |