Normalized defining polynomial
\( x^{41} + 4x - 4 \)
Invariants
Degree: | $41$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[1, 20]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(3438740475870829672142694556791350302292427122659925573655422607717853822976\) \(\medspace = 2^{40}\cdot 63506090654183\cdot 49\!\cdots\!47\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(69.56\) | 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: | $2^{40/41}63506090654183^{1/2}49247500694181610257024215345813183672691438654247^{1/2}\approx 1.0997338366416402e+32$ | ||
Ramified primes: | \(2\), \(63506090654183\), \(49247\!\cdots\!54247\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{31275\!\cdots\!65201}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{2}a^{21}$, $\frac{1}{2}a^{22}$, $\frac{1}{2}a^{23}$, $\frac{1}{2}a^{24}$, $\frac{1}{2}a^{25}$, $\frac{1}{2}a^{26}$, $\frac{1}{2}a^{27}$, $\frac{1}{2}a^{28}$, $\frac{1}{2}a^{29}$, $\frac{1}{2}a^{30}$, $\frac{1}{2}a^{31}$, $\frac{1}{2}a^{32}$, $\frac{1}{2}a^{33}$, $\frac{1}{2}a^{34}$, $\frac{1}{2}a^{35}$, $\frac{1}{2}a^{36}$, $\frac{1}{2}a^{37}$, $\frac{1}{2}a^{38}$, $\frac{1}{2}a^{39}$, $\frac{1}{42}a^{40}-\frac{5}{21}a^{39}-\frac{5}{42}a^{38}+\frac{4}{21}a^{37}+\frac{2}{21}a^{36}+\frac{1}{21}a^{35}+\frac{1}{42}a^{34}-\frac{5}{21}a^{33}-\frac{5}{42}a^{32}+\frac{4}{21}a^{31}+\frac{2}{21}a^{30}+\frac{1}{21}a^{29}+\frac{1}{42}a^{28}-\frac{5}{21}a^{27}-\frac{5}{42}a^{26}+\frac{4}{21}a^{25}+\frac{2}{21}a^{24}+\frac{1}{21}a^{23}+\frac{1}{42}a^{22}-\frac{5}{21}a^{21}+\frac{8}{21}a^{20}+\frac{4}{21}a^{19}+\frac{2}{21}a^{18}+\frac{1}{21}a^{17}-\frac{10}{21}a^{16}-\frac{5}{21}a^{15}+\frac{8}{21}a^{14}+\frac{4}{21}a^{13}+\frac{2}{21}a^{12}+\frac{1}{21}a^{11}-\frac{10}{21}a^{10}-\frac{5}{21}a^{9}+\frac{8}{21}a^{8}+\frac{4}{21}a^{7}+\frac{2}{21}a^{6}+\frac{1}{21}a^{5}-\frac{10}{21}a^{4}-\frac{5}{21}a^{3}+\frac{8}{21}a^{2}+\frac{4}{21}a+\frac{4}{21}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
not computed
Unit group
Rank: | $20$ | 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 33452526613163807108170062053440751665152000000000 |
The 44583 conjugacy class representatives for $S_{41}$ are not computed |
Character table for $S_{41}$ 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 | $39{,}\,{\href{/padicField/3.2.0.1}{2} }$ | $31{,}\,{\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | $22{,}\,{\href{/padicField/7.14.0.1}{14} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | $30{,}\,{\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.13.0.1}{13} }{,}\,{\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.11.0.1}{11} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | $19{,}\,18{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $37{,}\,{\href{/padicField/19.4.0.1}{4} }$ | $36{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | $24{,}\,17$ | $20{,}\,{\href{/padicField/31.14.0.1}{14} }{,}\,{\href{/padicField/31.7.0.1}{7} }$ | $19{,}\,{\href{/padicField/37.11.0.1}{11} }{,}\,{\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.5.0.1}{5} }^{8}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $34{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | $23{,}\,{\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.8.0.1}{8} }$ | $21{,}\,{\href{/padicField/59.8.0.1}{8} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $41$ | $41$ | $1$ | $40$ | |||
\(63506090654183\) | $\Q_{63506090654183}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $32$ | $1$ | $32$ | $0$ | 32T33 | $[\ ]^{32}$ | ||
\(492\!\cdots\!247\) | $\Q_{49\!\cdots\!47}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ |