Normalized defining polynomial
\( x^{41} - x - 4 \)
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: | \(1463315430027639250498776006138655422510960361093027177982041370003573535932416\) \(\medspace = 2^{40}\cdot 13\!\cdots\!41\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(80.63\) | 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: | \(2\), \(13308\!\cdots\!53641\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{13308\!\cdots\!53641}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{2}a^{21}-\frac{1}{2}a$, $\frac{1}{2}a^{22}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{23}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{24}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{25}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{26}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{27}-\frac{1}{2}a^{7}$, $\frac{1}{2}a^{28}-\frac{1}{2}a^{8}$, $\frac{1}{2}a^{29}-\frac{1}{2}a^{9}$, $\frac{1}{2}a^{30}-\frac{1}{2}a^{10}$, $\frac{1}{2}a^{31}-\frac{1}{2}a^{11}$, $\frac{1}{2}a^{32}-\frac{1}{2}a^{12}$, $\frac{1}{2}a^{33}-\frac{1}{2}a^{13}$, $\frac{1}{2}a^{34}-\frac{1}{2}a^{14}$, $\frac{1}{2}a^{35}-\frac{1}{2}a^{15}$, $\frac{1}{2}a^{36}-\frac{1}{2}a^{16}$, $\frac{1}{2}a^{37}-\frac{1}{2}a^{17}$, $\frac{1}{2}a^{38}-\frac{1}{2}a^{18}$, $\frac{1}{2}a^{39}-\frac{1}{2}a^{19}$, $\frac{1}{2}a^{40}-\frac{1}{2}a^{20}$
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 | $41$ | $31{,}\,{\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | $28{,}\,{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.6.0.1}{6} }$ | $39{,}\,{\href{/padicField/11.2.0.1}{2} }$ | $19{,}\,17{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | $29{,}\,{\href{/padicField/17.12.0.1}{12} }$ | $15{,}\,{\href{/padicField/19.13.0.1}{13} }{,}\,{\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | $27{,}\,{\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | $23{,}\,16{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | $23{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ | $34{,}\,{\href{/padicField/37.7.0.1}{7} }$ | $41$ | $28{,}\,{\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.6.0.1}{6} }$ | $15{,}\,{\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | $38{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/59.13.0.1}{13} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
2.8.8.11 | $x^{8} + 4 x^{7} + 14 x^{6} + 32 x^{5} + 55 x^{4} + 60 x^{3} + 36 x^{2} + 18 x + 9$ | $4$ | $2$ | $8$ | $S_4$ | $[4/3, 4/3]_{3}^{2}$ | |
Deg $32$ | $4$ | $8$ | $32$ | ||||
\(133\!\cdots\!641\) | $\Q_{13\!\cdots\!41}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $36$ | $1$ | $36$ | $0$ | $C_{36}$ | $[\ ]^{36}$ |