Normalized defining polynomial
\( x^{41} + 5x - 5 \)
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: | \(126\!\cdots\!625\) \(\medspace = 5^{40}\cdot 19\cdot 607\cdot 1420817\cdot 36720491\cdot 23\!\cdots\!91\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(197.32\) | 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: | $5^{40/41}19^{1/2}607^{1/2}1420817^{1/2}36720491^{1/2}2312274212437627784352237135454468699991228948791^{1/2}\approx 5.670692287910047e+33$ | ||
Ramified primes: | \(5\), \(19\), \(607\), \(1420817\), \(36720491\), \(23122\!\cdots\!48791\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{13913\!\cdots\!53641}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$
Monogenic: | Yes | |
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 $S_{41}$ |
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 | $32{,}\,{\href{/padicField/2.7.0.1}{7} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ | $41$ | R | $28{,}\,{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.6.0.1}{6} }$ | $40{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $22{,}\,19$ | $17{,}\,{\href{/padicField/17.13.0.1}{13} }{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | R | $18{,}\,16{,}\,{\href{/padicField/23.7.0.1}{7} }$ | $25{,}\,{\href{/padicField/29.14.0.1}{14} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | $22{,}\,{\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.4.0.1}{4} }$ | $38{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | $20^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | $27{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | $27{,}\,{\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $28{,}\,{\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | Deg $41$ | $41$ | $1$ | $40$ | |||
\(19\) | 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
19.3.0.1 | $x^{3} + 4 x + 17$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
Deg $36$ | $1$ | $36$ | $0$ | $C_{36}$ | $[\ ]^{36}$ | ||
\(607\) | $\Q_{607}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{607}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{607}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $34$ | $1$ | $34$ | $0$ | $C_{34}$ | $[\ ]^{34}$ | ||
\(1420817\) | 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 $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
\(36720491\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | ||
Deg $21$ | $1$ | $21$ | $0$ | $C_{21}$ | $[\ ]^{21}$ | ||
\(231\!\cdots\!791\) | $\Q_{23\!\cdots\!91}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{23\!\cdots\!91}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $34$ | $1$ | $34$ | $0$ | $C_{34}$ | $[\ ]^{34}$ |