Normalized defining polynomial
\( x^{23} + 9x - 9 \)
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: | \(-751680528112712572094133652841196244056207\) \(\medspace = -\,3^{22}\cdot 151\cdot 158631250967721112920032141873\) | 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: | \(66.18\) | 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))
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Galois root discriminant: | $3^{22/23}151^{1/2}158631250967721112920032141873^{1/2}\approx 1.3997797308565002e+16$ | ||
Ramified primes: | \(3\), \(151\), \(158631250967721112920032141873\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | $\Q(\sqrt{-23953\!\cdots\!22823}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3}a^{12}$, $\frac{1}{3}a^{13}$, $\frac{1}{3}a^{14}$, $\frac{1}{3}a^{15}$, $\frac{1}{3}a^{16}$, $\frac{1}{3}a^{17}$, $\frac{1}{3}a^{18}$, $\frac{1}{3}a^{19}$, $\frac{1}{3}a^{20}$, $\frac{1}{3}a^{21}$, $\frac{1}{3}a^{22}$
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)
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Fundamental units: | $a-1$, $\frac{1}{3}a^{12}-a^{2}+1$, $\frac{1}{3}a^{21}+\frac{1}{3}a^{20}+\frac{1}{3}a^{19}+\frac{1}{3}a^{18}+\frac{1}{3}a^{17}-\frac{1}{3}a^{14}-\frac{2}{3}a^{13}-\frac{2}{3}a^{12}+a^{5}+2a^{4}+a^{3}-a-2$, $\frac{2}{3}a^{22}+\frac{2}{3}a^{21}+\frac{1}{3}a^{20}+\frac{1}{3}a^{17}+\frac{2}{3}a^{16}+\frac{2}{3}a^{15}+\frac{1}{3}a^{14}+\frac{1}{3}a^{13}+\frac{1}{3}a^{12}-a^{10}-a^{9}+a^{7}+a^{6}-a^{4}-a^{3}-a+4$, $\frac{1}{3}a^{21}-\frac{1}{3}a^{20}-\frac{2}{3}a^{18}-\frac{1}{3}a^{17}+\frac{1}{3}a^{16}-\frac{1}{3}a^{15}+\frac{4}{3}a^{14}-\frac{1}{3}a^{13}+\frac{1}{3}a^{12}-a^{11}-a^{10}-a^{9}-a^{8}+2a^{7}-a^{6}+4a^{5}-a^{4}-4a+2$, $\frac{2}{3}a^{22}-\frac{4}{3}a^{21}-\frac{2}{3}a^{20}+\frac{5}{3}a^{19}+a^{18}-\frac{7}{3}a^{17}-\frac{2}{3}a^{16}+\frac{8}{3}a^{15}+a^{14}-\frac{11}{3}a^{13}-a^{12}+5a^{11}-5a^{9}-a^{8}+8a^{7}-10a^{5}+2a^{4}+10a^{3}-16a+10$, $\frac{2}{3}a^{22}-\frac{1}{3}a^{20}-\frac{1}{3}a^{19}-\frac{1}{3}a^{17}-\frac{4}{3}a^{16}-\frac{4}{3}a^{15}+a^{13}+\frac{2}{3}a^{12}+2a^{9}+3a^{8}-a^{7}-3a^{6}-a^{5}+a^{3}-3a^{2}-6a+8$, $\frac{1}{3}a^{22}-\frac{2}{3}a^{21}-a^{20}+\frac{1}{3}a^{19}-a^{18}-\frac{2}{3}a^{17}+\frac{5}{3}a^{16}-\frac{1}{3}a^{15}+\frac{1}{3}a^{14}+\frac{10}{3}a^{13}-\frac{2}{3}a^{12}+3a^{10}-3a^{9}-2a^{8}+2a^{7}-5a^{6}-a^{5}+3a^{4}-4a^{3}+4a^{2}+5a-2$, $\frac{1}{3}a^{22}+\frac{2}{3}a^{21}+\frac{1}{3}a^{20}+a^{19}+\frac{7}{3}a^{18}+\frac{7}{3}a^{17}+\frac{5}{3}a^{16}+\frac{7}{3}a^{15}+\frac{11}{3}a^{14}+3a^{13}+a^{12}+2a^{11}+4a^{10}+2a^{9}-a^{8}+2a^{6}-a^{5}-5a^{4}-2a^{3}+2a^{2}-4a-7$, $\frac{31}{3}a^{22}+\frac{31}{3}a^{21}+10a^{20}+9a^{19}+\frac{26}{3}a^{18}+\frac{26}{3}a^{17}+\frac{25}{3}a^{16}+8a^{15}+\frac{23}{3}a^{14}+\frac{19}{3}a^{13}+7a^{12}+8a^{11}+5a^{10}+6a^{9}+7a^{8}+4a^{7}+7a^{6}+5a^{5}+2a^{4}+8a^{3}+4a^{2}+3a+100$, $a^{22}-\frac{4}{3}a^{19}-3a^{18}-\frac{2}{3}a^{17}+\frac{13}{3}a^{16}+5a^{15}+\frac{5}{3}a^{14}+a^{13}+\frac{5}{3}a^{12}-3a^{11}-8a^{10}-3a^{9}+7a^{8}+7a^{7}+a^{6}+3a^{5}+5a^{4}-9a^{3}-21a^{2}-7a+22$ (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)]
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Regulator: | \( 640542907885 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{11}\cdot 640542907885 \cdot 1}{2\cdot\sqrt{751680528112712572094133652841196244056207}}\cr\approx \mathstrut & 0.445153844924392 \end{aligned}\] (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 | ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | $20{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.11.0.1}{11} }{,}\,{\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.11.0.1}{11} }{,}\,{\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.11.0.1}{11} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ | $22{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | $15{,}\,{\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 | $x^{23} + 3$ | $23$ | $1$ | $22$ | $C_{23}:C_{11}$ | $[\ ]_{23}^{11}$ |
\(151\) | 151.2.1.2 | $x^{2} + 151$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
151.2.0.1 | $x^{2} + 149 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
151.5.0.1 | $x^{5} + 11 x + 145$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
151.14.0.1 | $x^{14} - x + 6$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | |
\(158\!\cdots\!873\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ |