Normalized defining polynomial
\( x^{23} + 3x - 3 \)
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)
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Discriminant: | \(-687394316682267077687567046704171943344271\) \(\medspace = -\,3^{22}\cdot 7\cdot 467399\cdot 10931299891\cdot 612464196024013\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(65.92\) | 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: | $3^{22/23}7^{1/2}467399^{1/2}10931299891^{1/2}612464196024013^{1/2}\approx 1.3385852001185396e+16$ | ||
Ramified primes: | \(3\), \(7\), \(467399\), \(10931299891\), \(612464196024013\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-21904\!\cdots\!81319}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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)
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Fundamental units: | $a-1$, $a^{6}+a^{5}-a^{3}-a^{2}+1$, $a^{17}-a^{15}+a^{13}-a^{11}+a^{10}+a^{7}-a^{6}-2a^{5}+a^{4}+3a^{3}-2a^{2}-a+1$, $a^{22}+a^{21}-a^{20}-3a^{19}-a^{18}+a^{17}+2a^{16}+3a^{15}-a^{14}-3a^{13}+a^{11}+4a^{10}+4a^{9}-2a^{8}-3a^{7}-a^{6}+a^{5}+5a^{4}+4a^{3}-3a^{2}-5a+1$, $a^{22}-2a^{21}-5a^{20}-6a^{19}-4a^{18}+a^{17}+5a^{16}+6a^{15}+3a^{14}-a^{13}-6a^{12}-8a^{11}-6a^{10}+a^{9}+7a^{8}+8a^{7}+5a^{6}-6a^{4}-11a^{3}-8a^{2}+11$, $a^{22}+3a^{21}+2a^{20}-3a^{19}-a^{18}-a^{17}-4a^{16}+a^{15}-4a^{13}+3a^{12}+2a^{11}+8a^{9}+a^{8}-4a^{7}+3a^{6}-4a^{5}-5a^{4}+6a^{3}-6a^{2}-6a+10$, $2a^{22}+2a^{21}+2a^{20}-a^{19}-3a^{18}-a^{17}-a^{15}-a^{14}-2a^{13}-5a^{12}-4a^{11}+a^{10}+a^{9}-2a^{8}-a^{7}-2a^{6}-4a^{5}+2a^{4}+7a^{3}+a^{2}-3a+7$, $5a^{22}-6a^{21}+5a^{20}+a^{19}-4a^{18}+10a^{17}-7a^{16}+5a^{15}+6a^{14}-9a^{13}+14a^{12}-7a^{11}+a^{10}+11a^{9}-17a^{8}+17a^{7}-7a^{6}-9a^{5}+21a^{4}-26a^{3}+16a^{2}-8$, $a^{22}+2a^{21}+a^{19}-a^{18}+a^{17}-2a^{13}+2a^{12}+a^{11}+6a^{10}+3a^{9}+3a^{8}+2a^{7}-a^{6}+3a^{5}-3a^{4}+2a^{3}-6a^{2}-4a-1$, $11a^{22}+6a^{21}+4a^{20}+6a^{19}+10a^{18}+13a^{17}+8a^{16}+2a^{15}-2a^{14}+3a^{13}+11a^{12}+12a^{11}+6a^{10}-4a^{9}-4a^{8}+2a^{7}+11a^{6}+11a^{5}+4a^{4}-4a^{3}-8a^{2}+a+40$, $7a^{22}+2a^{21}+10a^{20}-2a^{19}+10a^{18}-a^{17}+7a^{16}+4a^{15}-2a^{14}+10a^{13}-7a^{12}+15a^{11}-9a^{10}+8a^{9}-a^{7}+13a^{6}-17a^{5}+21a^{4}-18a^{3}+19a^{2}-8a+19$ (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: | \( 993709440615 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{11}\cdot 993709440615 \cdot 1}{2\cdot\sqrt{687394316682267077687567046704171943344271}}\cr\approx \mathstrut & 0.722162610216076 \end{aligned}\] (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 | ${\href{/padicField/2.13.0.1}{13} }{,}\,{\href{/padicField/2.8.0.1}{8} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ | R | $15{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | R | $20{,}\,{\href{/padicField/11.1.0.1}{1} }^{3}$ | ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | $22{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | $20{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.9.0.1}{9} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | $17{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.10.0.1}{10} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.23.22.1 | $x^{23} + 3$ | $23$ | $1$ | $22$ | $C_{23}:C_{11}$ | $[\ ]_{23}^{11}$ |
\(7\) | $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
7.17.0.1 | $x^{17} + x + 4$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | |
\(467399\) | $\Q_{467399}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
\(10931299891\) | $\Q_{10931299891}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{10931299891}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
\(612464196024013\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ |