Normalized defining polynomial
\( x^{23} + 4x - 1 \)
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: | \(-24025850957386483666274491107673457027738343\) \(\medspace = -\,3\cdot 307\cdot 34500939701\cdot 12135377781337\cdot 62306724912060059\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(76.93\) | 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: | $3^{1/2}307^{1/2}34500939701^{1/2}12135377781337^{1/2}62306724912060059^{1/2}\approx 4.901617177767607e+21$ | ||
Ramified primes: | \(3\), \(307\), \(34500939701\), \(12135377781337\), \(62306724912060059\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-24025\!\cdots\!38343}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$ (assuming GRH)
Unit group
Rank: | $11$ | 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: | $a^{22}+4$, $a^{22}+a^{19}-a^{18}+a^{17}-a^{15}+a^{14}-a^{13}-a^{10}+a^{9}-a^{8}+2a^{6}-2a^{5}+2a^{4}+a^{3}-2a^{2}+3a+1$, $a^{22}-a^{21}+2a^{20}-3a^{19}+3a^{18}-a^{17}-a^{16}+a^{15}-a^{14}+3a^{13}-5a^{12}+5a^{11}-3a^{10}+2a^{8}-2a^{7}+3a^{6}-7a^{5}+10a^{4}-7a^{3}+a^{2}+2a+1$, $a^{21}+2a^{20}+2a^{19}+a^{18}+a^{15}+2a^{14}+2a^{13}+a^{12}+a^{9}+2a^{8}+2a^{7}+a^{6}+a^{3}+2a^{2}+2a$, $a^{22}-a^{17}+a^{15}+a^{14}-a^{8}+a^{6}+2a^{5}-a^{2}+2$, $2a^{22}-3a^{21}+3a^{20}-4a^{19}+4a^{18}-4a^{17}+4a^{16}-4a^{15}+3a^{14}-3a^{13}+a^{12}-2a^{10}+4a^{9}-4a^{8}+6a^{7}-7a^{6}+7a^{5}-9a^{4}+10a^{3}-9a^{2}+11a-3$, $8a^{22}-6a^{21}-4a^{20}+10a^{19}-2a^{18}-11a^{17}+7a^{16}+8a^{15}-12a^{14}-a^{13}+16a^{12}-10a^{11}-13a^{10}+19a^{9}+2a^{8}-21a^{7}+11a^{6}+16a^{5}-24a^{4}-8a^{3}+33a^{2}-9a+2$, $8a^{21}-4a^{20}-7a^{19}+8a^{18}+3a^{17}-5a^{16}+a^{15}-a^{14}+4a^{13}+4a^{12}-13a^{11}+5a^{10}+17a^{9}-16a^{8}-8a^{7}+16a^{6}-2a^{5}-3a^{4}-2a^{3}-7a^{2}+18a-3$, $4a^{22}+8a^{21}+10a^{20}+10a^{19}+10a^{18}+10a^{17}+11a^{16}+14a^{15}+17a^{14}+18a^{13}+14a^{12}+10a^{11}+11a^{10}+13a^{9}+14a^{8}+11a^{7}+6a^{6}+a^{5}-6a^{4}-6a^{3}-3a^{2}-5a+1$, $2a^{22}-3a^{21}+a^{20}-a^{19}+a^{18}+4a^{17}-a^{16}+6a^{15}-4a^{14}-a^{13}-10a^{12}-3a^{11}+6a^{10}+4a^{9}+13a^{8}-5a^{7}+4a^{6}-15a^{5}-7a^{4}+a^{3}+a^{2}+17a-3$, $2a^{22}+25a^{21}+12a^{20}-19a^{19}-23a^{18}+8a^{17}+30a^{16}+12a^{15}-26a^{14}-28a^{13}+12a^{12}+43a^{11}+13a^{10}-42a^{9}-40a^{8}+29a^{7}+63a^{6}+2a^{5}-73a^{4}-40a^{3}+61a^{2}+74a-20$ (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: | \( 2707199797350 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{11}\cdot 2707199797350 \cdot 2}{2\cdot\sqrt{24025850957386483666274491107673457027738343}}\cr\approx \mathstrut & 0.665563454384698 \end{aligned}\] (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 | ${\href{/padicField/2.11.0.1}{11} }^{2}{,}\,{\href{/padicField/2.1.0.1}{1} }$ | R | ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | $20{,}\,{\href{/padicField/11.3.0.1}{3} }$ | $23$ | ${\href{/padicField/17.11.0.1}{11} }{,}\,{\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/31.4.0.1}{4} }$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | $18{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.6.0.1}{6} }$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.6.0.1}{6} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.3.0.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | ||
\(307\) | $\Q_{307}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | ||
\(34500939701\) | $\Q_{34500939701}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{34500939701}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
\(12135377781337\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
\(62306724912060059\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ |