Normalized defining polynomial
\( x^{23} + 3x - 1 \)
Invariants
Degree: | $23$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[1, 11]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-32143105736103215203131215102867183266535\) \(\medspace = -\,5\cdot 22723592653\cdot 282905139402414328282680249319\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(57.70\) | 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: | $5^{1/2}22723592653^{1/2}282905139402414328282680249319^{1/2}\approx 1.7928498469225808e+20$ | ||
Ramified primes: | \(5\), \(22723592653\), \(282905139402414328282680249319\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-32143\!\cdots\!66535}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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)
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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: | $a^{22}+3$, $a^{21}+a^{18}-a^{17}-a^{16}+a^{15}-a^{14}+a^{12}-a^{11}+2a^{9}-2a^{8}+a^{6}-a^{5}+a^{4}+2a^{3}-3a^{2}+a+1$, $a^{22}-a^{21}+a^{20}-a^{18}+a^{17}-a^{15}+a^{14}-a^{12}+a^{11}-a^{9}+a^{8}-a^{6}+a^{5}-a^{3}+a^{2}+1$, $a^{22}+a^{21}-a^{19}-2a^{18}-a^{17}+a^{15}+2a^{14}+a^{13}-a^{11}-a^{10}-a^{9}+2a^{5}+a^{4}+a^{3}-2a$, $a^{16}-a^{13}+a^{12}-2a^{9}+a^{8}+a^{6}-2a^{5}+a^{4}+2a^{2}-a+1$, $2a^{22}-a^{21}-4a^{20}-3a^{19}+a^{18}+5a^{17}+3a^{16}-a^{15}-6a^{14}-3a^{13}+a^{12}+6a^{11}+3a^{10}-6a^{8}-3a^{7}-a^{6}+5a^{5}+5a^{4}+3a^{3}-5a^{2}-7a+2$, $2a^{22}-a^{21}-a^{20}-a^{19}+2a^{17}+2a^{16}-a^{15}-a^{14}-2a^{13}+2a^{11}+3a^{10}-a^{9}-a^{8}-3a^{7}-a^{6}+3a^{5}+3a^{4}-a^{2}-4a+3$, $a^{22}-a^{21}+2a^{20}-3a^{18}+2a^{17}-3a^{15}+3a^{14}-a^{13}-4a^{12}+5a^{11}+a^{10}-4a^{9}+4a^{8}-3a^{6}+5a^{5}-2a^{4}-4a^{3}+4a^{2}-3a+2$, $a^{17}-a^{16}+a^{15}-a^{14}+a^{13}-a^{12}+a^{9}-a^{8}+a^{7}-a^{6}+a^{5}-a^{4}-a^{3}+a^{2}+1$, $4a^{22}-5a^{21}+4a^{20}-2a^{19}+a^{18}+a^{17}-4a^{16}+5a^{15}-5a^{14}+5a^{13}-4a^{12}+3a^{10}-5a^{9}+8a^{8}-10a^{7}+8a^{6}-5a^{5}+3a^{4}+3a^{3}-10a^{2}+13a-4$, $3a^{22}+4a^{21}+2a^{20}+3a^{19}+6a^{18}+3a^{17}+a^{16}+4a^{15}+3a^{14}-a^{13}+2a^{12}+4a^{11}-a^{10}-a^{9}+3a^{8}-2a^{7}-7a^{6}-a^{5}-2a^{4}-8a^{3}-5a^{2}+a$ (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: | \( 185847611513 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{11}\cdot 185847611513 \cdot 1}{2\cdot\sqrt{32143105736103215203131215102867183266535}}\cr\approx \mathstrut & 0.624585096972757 \end{aligned}\] (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 | ${\href{/padicField/2.13.0.1}{13} }{,}\,{\href{/padicField/2.8.0.1}{8} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ | ${\href{/padicField/3.11.0.1}{11} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }$ | R | $20{,}\,{\href{/padicField/7.3.0.1}{3} }$ | $22{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.11.0.1}{11} }{,}\,{\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/19.3.0.1}{3} }$ | $22{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $23$ | $16{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.9.0.1}{9} }{,}\,{\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $23$ | $17{,}\,{\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | $22{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | 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 | $x^{12} + x^{7} + x^{6} + 4 x^{4} + 4 x^{3} + 3 x^{2} + 2 x + 2$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
\(22723592653\) | $\Q_{22723592653}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
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 $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
\(282\!\cdots\!319\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |