Normalized defining polynomial
\( x^{43} + 3x - 2 \)
Invariants
Degree: | $43$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[1, 21]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-492\!\cdots\!856\) \(\medspace = -\,2^{43}\cdot 83\cdot 67\!\cdots\!29\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(115.51\) | 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: | \(2\), \(83\), \(67502\!\cdots\!34329\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-11205\!\cdots\!98614}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $a^{42}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
not computed
Unit group
Rank: | $21$ | 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 60415263063373835637355132068513997507264512000000000 |
The 63261 conjugacy class representatives for $S_{43}$ are not computed |
Character table for $S_{43}$ 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 | $42{,}\,{\href{/padicField/3.1.0.1}{1} }$ | $43$ | ${\href{/padicField/7.13.0.1}{13} }{,}\,{\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $41{,}\,{\href{/padicField/11.2.0.1}{2} }$ | $22{,}\,15{,}\,{\href{/padicField/13.6.0.1}{6} }$ | $16{,}\,{\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | $29{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.4.0.1}{4} }^{2}$ | $34{,}\,{\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.4.0.1}{4} }$ | $27{,}\,{\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}$ | $27{,}\,{\href{/padicField/31.13.0.1}{13} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | $24{,}\,18{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $21^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}$ | $35{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | $15{,}\,{\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
2.2.3.2 | $x^{2} + 4 x + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
2.6.6.2 | $x^{6} + 6 x^{5} + 14 x^{4} + 24 x^{3} + 44 x^{2} + 8 x + 72$ | $2$ | $3$ | $6$ | $A_4\times C_2$ | $[2, 2]^{6}$ | |
2.6.6.2 | $x^{6} + 6 x^{5} + 14 x^{4} + 24 x^{3} + 44 x^{2} + 8 x + 72$ | $2$ | $3$ | $6$ | $A_4\times C_2$ | $[2, 2]^{6}$ | |
2.12.12.4 | $x^{12} + 6 x^{11} + 24 x^{10} + 84 x^{9} + 196 x^{8} + 336 x^{7} + 768 x^{6} + 2144 x^{5} + 4784 x^{4} + 8032 x^{3} + 9600 x^{2} + 6976 x + 2240$ | $2$ | $6$ | $12$ | 12T87 | $[2, 2, 2, 2, 2]^{6}$ | |
2.12.12.4 | $x^{12} + 6 x^{11} + 24 x^{10} + 84 x^{9} + 196 x^{8} + 336 x^{7} + 768 x^{6} + 2144 x^{5} + 4784 x^{4} + 8032 x^{3} + 9600 x^{2} + 6976 x + 2240$ | $2$ | $6$ | $12$ | 12T87 | $[2, 2, 2, 2, 2]^{6}$ | |
\(83\) | 83.2.1.2 | $x^{2} + 83$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
83.2.0.1 | $x^{2} + 82 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
83.2.0.1 | $x^{2} + 82 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
83.3.0.1 | $x^{3} + 3 x + 81$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
83.5.0.1 | $x^{5} + 9 x + 81$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
83.5.0.1 | $x^{5} + 9 x + 81$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
83.8.0.1 | $x^{8} + x^{4} + 65 x^{3} + 23 x^{2} + 42 x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
83.16.0.1 | $x^{16} - x + 73$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | |
\(675\!\cdots\!329\) | $\Q_{67\!\cdots\!29}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
Deg $24$ | $1$ | $24$ | $0$ | $C_{24}$ | $[\ ]^{24}$ |