Normalized defining polynomial
\( x^{23} - 3x - 3 \)
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: | \(-623108105251821583281000440567147642632335\) \(\medspace = -\,3^{22}\cdot 5\cdot 203008301\cdot 19561943299801571525063\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(65.64\) | 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: | $3^{22/23}5^{1/2}203008301^{1/2}19561943299801571525063^{1/2}\approx 1.274455719527393e+16$ | ||
Ramified primes: | \(3\), \(5\), \(203008301\), \(19561943299801571525063\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-19856\!\cdots\!39815}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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
Trivial group, which has order $1$ (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: | $a+1$, $a^{22}-a^{21}+a^{20}-a^{19}+a^{18}-a^{17}+a^{16}-a^{15}+a^{14}-a^{13}+a^{12}-a^{11}+a^{10}-a^{9}+a^{8}-a^{7}+a^{6}+a^{2}-a-2$, $a^{22}-a^{20}+a^{19}-a^{18}+a^{16}-a^{15}+a^{13}-a^{12}+a^{11}+a^{10}-2a^{9}+a^{8}-a^{7}-a^{6}+2a^{5}-a^{4}-a^{3}+2a^{2}-a-2$, $a^{21}+a^{17}+a^{14}+a^{13}+a^{10}+a^{9}+a^{7}+2a^{6}+a^{5}-a^{4}+a^{3}+2a^{2}+a+1$, $a^{22}-a^{21}+a^{19}-a^{18}+a^{17}-a^{16}+a^{14}-a^{13}+a^{12}-a^{11}+a^{9}-a^{8}+a^{7}-a^{6}+a^{4}-a^{3}+a^{2}-a-2$, $a^{22}-a^{21}+a^{19}-a^{18}+a^{17}+a^{16}-a^{15}+a^{14}-a^{12}-2a^{9}+2a^{8}-a^{6}+4a^{5}-2a^{4}+a^{3}+2a^{2}-4a-2$, $a^{22}+a^{21}+a^{20}+a^{19}-2a^{17}-a^{16}-a^{15}+2a^{13}+3a^{12}-a^{11}-2a^{10}-2a^{9}-a^{8}+4a^{6}+3a^{5}+a^{4}-a^{3}-5a^{2}-8a-4$, $a^{22}+a^{21}-a^{19}-3a^{18}-2a^{17}-2a^{16}+a^{15}+a^{14}+2a^{13}+2a^{12}+3a^{11}+3a^{10}+2a^{9}-2a^{8}-4a^{7}-7a^{6}-3a^{5}-a^{4}+3a^{3}+3a^{2}+4a+2$, $2a^{22}-2a^{21}-2a^{20}+3a^{19}+a^{18}-3a^{17}+3a^{15}-a^{13}+2a^{12}+4a^{11}-3a^{9}+a^{8}+3a^{7}-2a^{6}-4a^{5}+3a^{4}+3a^{3}-8a^{2}-8a-1$, $a^{21}+a^{19}-a^{17}+a^{16}+a^{15}+a^{14}+2a^{13}-a^{12}-a^{11}-2a^{9}-2a^{6}-2a^{4}-3a^{3}-a^{2}-2a-1$, $5a^{22}+3a^{21}-10a^{20}+12a^{19}-9a^{18}+3a^{17}+3a^{16}-9a^{15}+11a^{14}-12a^{13}+4a^{12}+5a^{11}-16a^{10}+19a^{9}-13a^{8}+2a^{7}+9a^{6}-17a^{5}+17a^{4}-18a^{3}+7a^{2}+2a-34$ (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: | \( 933328504105 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{11}\cdot 933328504105 \cdot 1}{2\cdot\sqrt{623108105251821583281000440567147642632335}}\cr\approx \mathstrut & 0.712412254264647 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 25852016738884976640000 |
The 1255 conjugacy class representatives for $S_{23}$ are not computed |
Character table for $S_{23}$ is not computed |
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 | ${\href{/padicField/2.13.0.1}{13} }{,}\,{\href{/padicField/2.8.0.1}{8} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ | R | R | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $23$ | $19{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.11.0.1}{11} }{,}\,{\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.11.0.1}{11} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/29.6.0.1}{6} }$ | ${\href{/padicField/31.13.0.1}{13} }{,}\,{\href{/padicField/31.10.0.1}{10} }$ | ${\href{/padicField/37.14.0.1}{14} }{,}\,{\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | $22{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.5.0.1}{5} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.14.0.1}{14} }{,}\,{\href{/padicField/59.9.0.1}{9} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.23.22.1 | $x^{23} + 3$ | $23$ | $1$ | $22$ | $C_{23}:C_{11}$ | $[\ ]_{23}^{11}$ |
\(5\) | $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.8.0.1 | $x^{8} + x^{4} + 3 x^{2} + 4 x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
5.12.0.1 | $x^{12} + x^{7} + x^{6} + 4 x^{4} + 4 x^{3} + 3 x^{2} + 2 x + 2$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
\(203008301\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
\(195\!\cdots\!063\) | $\Q_{19\!\cdots\!63}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ |