Normalized defining polynomial
\( x^{23} - 9x - 9 \)
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: | \(-332434202154298684636783958613993659679\) \(\medspace = -\,3^{22}\cdot 103\cdot 409\cdot 136973\cdot 1835872831815206461\) | 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: | \(47.30\) | 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: | $3^{22/23}103^{1/2}409^{1/2}136973^{1/2}1835872831815206461^{1/2}\approx 294371594383291.06$ | ||
Ramified primes: | \(3\), \(103\), \(409\), \(136973\), \(1835872831815206461\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-10593\!\cdots\!10231}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3}a^{12}$, $\frac{1}{3}a^{13}$, $\frac{1}{3}a^{14}$, $\frac{1}{3}a^{15}$, $\frac{1}{3}a^{16}$, $\frac{1}{3}a^{17}$, $\frac{1}{3}a^{18}$, $\frac{1}{3}a^{19}$, $\frac{1}{3}a^{20}$, $\frac{1}{3}a^{21}$, $\frac{1}{123}a^{22}+\frac{4}{41}a^{21}-\frac{20}{123}a^{20}+\frac{2}{41}a^{19}-\frac{10}{123}a^{18}+\frac{1}{41}a^{17}-\frac{5}{123}a^{16}-\frac{19}{123}a^{15}+\frac{6}{41}a^{14}+\frac{11}{123}a^{13}+\frac{3}{41}a^{12}-\frac{5}{41}a^{11}-\frac{19}{41}a^{10}+\frac{18}{41}a^{9}+\frac{11}{41}a^{8}+\frac{9}{41}a^{7}-\frac{15}{41}a^{6}-\frac{16}{41}a^{5}+\frac{13}{41}a^{4}-\frac{8}{41}a^{3}-\frac{14}{41}a^{2}-\frac{4}{41}a-\frac{10}{41}$
Monogenic: | Not computed | |
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: | $\frac{1}{3}a^{22}-\frac{1}{3}a^{21}+\frac{1}{3}a^{20}-\frac{1}{3}a^{19}+\frac{1}{3}a^{18}-\frac{1}{3}a^{17}+\frac{1}{3}a^{16}-\frac{1}{3}a^{15}+\frac{1}{3}a^{14}-\frac{1}{3}a^{13}+\frac{1}{3}a^{12}-2$, $\frac{1}{3}a^{22}-\frac{1}{3}a^{21}+\frac{1}{3}a^{20}-\frac{1}{3}a^{19}+\frac{1}{3}a^{18}-\frac{1}{3}a^{17}+\frac{1}{3}a^{16}-\frac{1}{3}a^{15}+\frac{1}{3}a^{14}-\frac{1}{3}a^{13}+\frac{1}{3}a^{12}-4$, $\frac{31}{123}a^{22}-\frac{38}{123}a^{21}+\frac{12}{41}a^{20}-\frac{19}{123}a^{19}-\frac{23}{123}a^{18}+\frac{52}{123}a^{17}-\frac{73}{123}a^{16}+\frac{67}{123}a^{15}-\frac{16}{123}a^{14}-\frac{28}{123}a^{13}+\frac{74}{123}a^{12}-\frac{32}{41}a^{11}+\frac{26}{41}a^{10}-\frac{16}{41}a^{9}-\frac{28}{41}a^{8}+\frac{33}{41}a^{7}-\frac{55}{41}a^{6}+\frac{37}{41}a^{5}-\frac{7}{41}a^{4}-\frac{2}{41}a^{3}+\frac{58}{41}a^{2}-\frac{42}{41}a-\frac{23}{41}$, $\frac{16}{123}a^{22}-\frac{13}{123}a^{21}+\frac{49}{123}a^{20}-\frac{9}{41}a^{19}+\frac{15}{41}a^{18}+\frac{7}{123}a^{17}+\frac{43}{123}a^{16}-\frac{17}{123}a^{15}+\frac{14}{41}a^{14}+\frac{4}{41}a^{13}-\frac{20}{123}a^{12}+\frac{2}{41}a^{11}-\frac{17}{41}a^{10}+\frac{1}{41}a^{9}-\frac{29}{41}a^{8}-\frac{20}{41}a^{7}-\frac{35}{41}a^{6}-\frac{10}{41}a^{5}-\frac{38}{41}a^{4}-\frac{46}{41}a^{3}+\frac{22}{41}a^{2}-\frac{23}{41}a-\frac{37}{41}$, $\frac{6}{41}a^{22}-\frac{10}{41}a^{21}+\frac{3}{41}a^{20}+\frac{26}{123}a^{19}-\frac{19}{41}a^{18}+\frac{18}{41}a^{17}-\frac{8}{123}a^{16}-\frac{55}{123}a^{15}+\frac{26}{41}a^{14}-\frac{16}{41}a^{13}-\frac{43}{123}a^{12}+\frac{33}{41}a^{11}-\frac{14}{41}a^{10}-\frac{4}{41}a^{9}+\frac{34}{41}a^{8}-\frac{43}{41}a^{7}+\frac{17}{41}a^{6}-\frac{1}{41}a^{5}-\frac{53}{41}a^{4}+\frac{61}{41}a^{3}-\frac{6}{41}a^{2}-\frac{31}{41}a+\frac{25}{41}$, $\frac{32}{41}a^{22}-\frac{119}{123}a^{21}+\frac{89}{123}a^{20}-\frac{13}{41}a^{19}+\frac{8}{41}a^{18}+\frac{1}{123}a^{17}-\frac{70}{123}a^{16}+\frac{48}{41}a^{15}-\frac{39}{41}a^{14}+\frac{113}{123}a^{13}-\frac{202}{123}a^{12}+\frac{53}{41}a^{11}-\frac{20}{41}a^{10}+\frac{6}{41}a^{9}-\frac{10}{41}a^{8}-\frac{38}{41}a^{7}+\frac{77}{41}a^{6}-\frac{60}{41}a^{5}+\frac{59}{41}a^{4}-\frac{112}{41}a^{3}+\frac{91}{41}a^{2}-\frac{15}{41}a-\frac{263}{41}$, $\frac{8}{41}a^{22}-\frac{40}{123}a^{21}+\frac{4}{41}a^{20}+\frac{7}{41}a^{19}-\frac{35}{123}a^{18}-\frac{10}{123}a^{17}+\frac{44}{123}a^{16}-\frac{46}{123}a^{15}+\frac{22}{123}a^{14}+\frac{6}{41}a^{13}+\frac{11}{123}a^{12}+\frac{3}{41}a^{11}-\frac{5}{41}a^{10}+\frac{22}{41}a^{9}-\frac{23}{41}a^{8}-\frac{30}{41}a^{7}+\frac{50}{41}a^{6}-\frac{56}{41}a^{5}-\frac{16}{41}a^{4}+\frac{95}{41}a^{3}-\frac{49}{41}a^{2}-\frac{14}{41}a+\frac{47}{41}$, $\frac{4}{123}a^{22}-\frac{34}{123}a^{21}+\frac{43}{123}a^{20}+\frac{8}{41}a^{19}-\frac{40}{123}a^{18}-\frac{29}{123}a^{17}+\frac{7}{41}a^{16}+\frac{2}{41}a^{15}+\frac{31}{123}a^{14}+\frac{44}{123}a^{13}-\frac{128}{123}a^{12}-\frac{20}{41}a^{11}+\frac{47}{41}a^{10}+\frac{31}{41}a^{9}-\frac{38}{41}a^{8}-\frac{5}{41}a^{7}-\frac{19}{41}a^{6}-\frac{23}{41}a^{5}+\frac{52}{41}a^{4}+\frac{50}{41}a^{3}-\frac{56}{41}a^{2}-\frac{16}{41}a+\frac{1}{41}$, $\frac{37}{123}a^{22}-\frac{16}{41}a^{21}+\frac{13}{41}a^{20}-\frac{8}{41}a^{19}-\frac{1}{123}a^{18}-\frac{53}{123}a^{17}+\frac{61}{123}a^{16}-\frac{47}{123}a^{15}+\frac{10}{123}a^{14}-\frac{1}{41}a^{13}+\frac{5}{123}a^{12}-\frac{21}{41}a^{11}+\frac{35}{41}a^{10}+\frac{10}{41}a^{9}-\frac{3}{41}a^{8}+\frac{5}{41}a^{7}+\frac{19}{41}a^{6}-\frac{18}{41}a^{5}+\frac{30}{41}a^{4}+\frac{32}{41}a^{3}-\frac{26}{41}a^{2}-\frac{25}{41}a-\frac{83}{41}$, $\frac{40}{123}a^{22}-\frac{53}{123}a^{21}-\frac{7}{41}a^{20}+\frac{35}{123}a^{19}-\frac{113}{123}a^{18}+\frac{40}{41}a^{17}-\frac{200}{123}a^{16}+\frac{142}{123}a^{15}-\frac{223}{123}a^{14}+\frac{51}{41}a^{13}-\frac{173}{123}a^{12}+\frac{5}{41}a^{11}-\frac{22}{41}a^{10}-\frac{59}{41}a^{9}+\frac{30}{41}a^{8}-\frac{91}{41}a^{7}+\frac{56}{41}a^{6}-\frac{148}{41}a^{5}+\frac{28}{41}a^{4}-\frac{115}{41}a^{3}-\frac{27}{41}a^{2}-\frac{37}{41}a-\frac{236}{41}$, $\frac{94}{123}a^{22}-\frac{143}{123}a^{21}+\frac{211}{123}a^{20}-\frac{215}{123}a^{19}+\frac{208}{123}a^{18}-\frac{70}{41}a^{17}+\frac{145}{123}a^{16}-\frac{35}{41}a^{15}-\frac{10}{41}a^{14}+\frac{44}{41}a^{13}-\frac{220}{123}a^{12}+\frac{104}{41}a^{11}-\frac{105}{41}a^{10}+\frac{134}{41}a^{9}-\frac{114}{41}a^{8}+\frac{108}{41}a^{7}-\frac{57}{41}a^{6}-\frac{28}{41}a^{5}+\frac{33}{41}a^{4}-\frac{137}{41}a^{3}+\frac{119}{41}a^{2}-\frac{171}{41}a-\frac{38}{41}$ (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: | \( 12917754140.4 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{11}\cdot 12917754140.4 \cdot 1}{2\cdot\sqrt{332434202154298684636783958613993659679}}\cr\approx \mathstrut & 0.426886790957 \end{aligned}\] (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 | $16{,}\,{\href{/padicField/5.5.0.1}{5} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | $20{,}\,{\href{/padicField/7.3.0.1}{3} }$ | ${\href{/padicField/11.7.0.1}{7} }^{2}{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.11.0.1}{11} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/29.4.0.1}{4} }$ | $18{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.7.0.1}{7} }^{2}{,}\,{\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | $18{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | $17{,}\,{\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.14.0.1}{14} }{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | $19{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.11.0.1}{11} }{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.23.22.1 | $x^{23} + 3$ | $23$ | $1$ | $22$ | $C_{23}:C_{11}$ | $[\ ]_{23}^{11}$ |
\(103\) | 103.2.1.2 | $x^{2} + 103$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
103.4.0.1 | $x^{4} + 2 x^{2} + 88 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
103.17.0.1 | $x^{17} + 102 x^{2} + 8 x + 98$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | |
\(409\) | $\Q_{409}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
\(136973\) | $\Q_{136973}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{136973}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{136973}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{136973}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
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 $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
\(1835872831815206461\) | $\Q_{1835872831815206461}$ | $x$ | $1$ | $1$ | $0$ | 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 $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ |