Normalized defining polynomial
\( x^{23} - x - 3 \)
Invariants
Degree: | $23$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[1, 11]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-655251210966702902606919524078263146264719\) \(\medspace = -\,41\cdot 263\cdot 455033\cdot 13\!\cdots\!21\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(65.78\) | 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: | $41^{1/2}263^{1/2}455033^{1/2}133544293000689084344982443848921^{1/2}\approx 8.094758865875517e+20$ | ||
Ramified primes: | \(41\), \(263\), \(455033\), \(13354\!\cdots\!48921\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-65525\!\cdots\!64719}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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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: | $a^{12}-a-1$, $a^{22}-a^{21}+a^{20}+a^{18}+a^{15}+a^{13}+a^{11}+a^{9}+a^{8}+a^{6}+3a^{4}-a^{3}+a^{2}+a+1$, $a^{21}+a^{20}-2a^{19}+a^{17}+a^{14}-2a^{13}-2a^{12}+2a^{11}+a^{10}-a^{9}-3a^{6}+2a^{5}+3a^{4}-a^{3}-3a^{2}+2a-1$, $a^{22}+2a^{20}-2a^{17}-a^{16}+2a^{14}+2a^{13}+a^{12}-a^{11}-2a^{10}-a^{9}+2a^{7}+a^{6}+2a^{5}-a^{4}+a^{3}-3a^{2}-4$, $a^{18}+a^{16}-a^{15}+a^{14}-a^{13}+a^{12}-2a^{11}+a^{10}-a^{9}+2a^{8}-2a^{7}+a^{6}-a^{5}+2a^{4}-2a^{3}-a^{2}-2a+1$, $a^{22}+a^{21}-a^{20}-2a^{19}+2a^{17}+a^{16}-a^{13}-3a^{12}+3a^{10}+2a^{9}-2a^{8}-a^{7}+a^{6}+a^{5}-2a^{4}+a^{2}-a-2$, $a^{21}-2a^{20}+2a^{19}-a^{18}+a^{15}-2a^{14}+2a^{13}-a^{12}-a^{11}+3a^{10}-4a^{9}+3a^{8}-a^{7}-a^{5}+3a^{4}-4a^{3}+3a^{2}-a-1$, $3a^{22}+2a^{21}+2a^{20}+2a^{19}+a^{17}-2a^{16}-a^{15}-4a^{14}-4a^{13}-5a^{12}-6a^{11}-4a^{10}-6a^{9}-2a^{8}-3a^{7}+a^{6}+2a^{5}+4a^{4}+7a^{3}+6a^{2}+10a+4$, $a^{22}+2a^{21}+a^{20}+a^{18}+2a^{17}+2a^{16}+a^{12}-2a^{10}-2a^{9}-a^{8}+a^{7}-2a^{6}-3a^{5}-2a^{4}+3a^{3}+2a^{2}-a-1$, $a^{22}+a^{19}-2a^{18}-4a^{17}-a^{16}+a^{15}+2a^{14}+7a^{13}+7a^{12}-a^{11}-5a^{10}-6a^{9}-9a^{8}-5a^{7}+5a^{6}+7a^{5}+6a^{4}+7a^{3}-8a-4$, $3a^{22}-a^{21}+a^{20}-4a^{18}+4a^{17}+2a^{16}-7a^{15}-a^{14}+a^{13}+a^{12}+4a^{11}-8a^{10}-7a^{9}+10a^{8}+4a^{7}-3a^{6}-a^{5}-5a^{4}+8a^{3}+11a^{2}-9a-4$ (assuming GRH) | 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: | \( 1198293682960 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{11}\cdot 1198293682960 \cdot 1}{2\cdot\sqrt{655251210966702902606919524078263146264719}}\cr\approx \mathstrut & 0.891944651621854 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 25852016738884976640000 |
The 1255 conjugacy class representatives for $S_{23}$ |
Character table for $S_{23}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 46 sibling: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.13.0.1}{13} }{,}\,{\href{/padicField/2.8.0.1}{8} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ | ${\href{/padicField/3.5.0.1}{5} }^{4}{,}\,{\href{/padicField/3.1.0.1}{1} }^{3}$ | $16{,}\,{\href{/padicField/5.5.0.1}{5} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | $20{,}\,{\href{/padicField/7.3.0.1}{3} }$ | $21{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.11.0.1}{11} }{,}\,{\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | $20{,}\,{\href{/padicField/19.3.0.1}{3} }$ | $23$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.9.0.1}{9} }{,}\,{\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | R | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.5.0.1}{5} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.11.0.1}{11} }{,}\,{\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.11.0.1}{11} }{,}\,{\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(41\) | 41.2.1.1 | $x^{2} + 41$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
41.3.0.1 | $x^{3} + x + 35$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
41.18.0.1 | $x^{18} + x^{11} + 7 x^{10} + 20 x^{9} + 23 x^{8} + 35 x^{7} + 38 x^{6} + 24 x^{5} + 12 x^{4} + 29 x^{3} + 10 x^{2} + 6 x + 6$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | |
\(263\) | 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 $17$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | ||
\(455033\) | $\Q_{455033}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{455033}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $17$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | ||
\(133\!\cdots\!921\) | $\Q_{13\!\cdots\!21}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $20$ | $1$ | $20$ | $0$ | 20T1 | $[\ ]^{20}$ |