Normalized defining polynomial
\( x^{23} - 4x - 4 \)
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: | \(-1670424922585164788805622697952280576\) \(\medspace = -\,2^{22}\cdot 8623\cdot 10045730659\cdot 4597557191821267\) | 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: | \(37.58\) | 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: | $2^{22/23}8623^{1/2}10045730659^{1/2}4597557191821267^{1/2}\approx 1224687438025823.0$ | ||
Ramified primes: | \(2\), \(8623\), \(10045730659\), \(4597557191821267\) | 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: | $\Q(\sqrt{-39826\!\cdots\!69719}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}$, $\frac{1}{2}a^{13}$, $\frac{1}{2}a^{14}$, $\frac{1}{2}a^{15}$, $\frac{1}{2}a^{16}$, $\frac{1}{2}a^{17}$, $\frac{1}{2}a^{18}$, $\frac{1}{2}a^{19}$, $\frac{1}{2}a^{20}$, $\frac{1}{2}a^{21}$, $\frac{1}{14}a^{22}-\frac{1}{7}a^{21}-\frac{3}{14}a^{20}-\frac{1}{14}a^{19}+\frac{1}{7}a^{18}+\frac{3}{14}a^{17}+\frac{1}{14}a^{16}-\frac{1}{7}a^{15}-\frac{3}{14}a^{14}-\frac{1}{14}a^{13}+\frac{1}{7}a^{12}-\frac{2}{7}a^{11}-\frac{3}{7}a^{10}-\frac{1}{7}a^{9}+\frac{2}{7}a^{8}+\frac{3}{7}a^{7}+\frac{1}{7}a^{6}-\frac{2}{7}a^{5}-\frac{3}{7}a^{4}-\frac{1}{7}a^{3}+\frac{2}{7}a^{2}+\frac{3}{7}a-\frac{1}{7}$
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: | $\frac{1}{2}a^{22}-\frac{1}{2}a^{21}+\frac{1}{2}a^{20}-\frac{1}{2}a^{19}+\frac{1}{2}a^{18}-\frac{1}{2}a^{17}+\frac{1}{2}a^{16}-\frac{1}{2}a^{15}+\frac{1}{2}a^{14}-\frac{1}{2}a^{13}+\frac{1}{2}a^{12}-1$, $a+1$, $\frac{1}{14}a^{22}-\frac{1}{7}a^{21}+\frac{2}{7}a^{20}-\frac{1}{14}a^{19}+\frac{1}{7}a^{18}-\frac{2}{7}a^{17}+\frac{1}{14}a^{16}-\frac{1}{7}a^{15}+\frac{2}{7}a^{14}-\frac{1}{14}a^{13}+\frac{1}{7}a^{12}-\frac{2}{7}a^{11}-\frac{3}{7}a^{10}-\frac{1}{7}a^{9}+\frac{2}{7}a^{8}+\frac{3}{7}a^{7}+\frac{1}{7}a^{6}-\frac{2}{7}a^{5}-\frac{3}{7}a^{4}-\frac{1}{7}a^{3}+\frac{2}{7}a^{2}+\frac{3}{7}a-\frac{1}{7}$, $\frac{1}{2}a^{22}-\frac{1}{2}a^{21}+\frac{1}{2}a^{20}-\frac{1}{2}a^{19}+\frac{1}{2}a^{18}-\frac{1}{2}a^{17}+\frac{1}{2}a^{16}-\frac{1}{2}a^{15}+\frac{1}{2}a^{14}-3$, $\frac{1}{7}a^{22}-\frac{2}{7}a^{21}+\frac{1}{14}a^{20}+\frac{5}{14}a^{19}-\frac{3}{14}a^{18}-\frac{1}{14}a^{17}+\frac{1}{7}a^{16}+\frac{3}{14}a^{15}-\frac{3}{7}a^{14}-\frac{1}{7}a^{13}+\frac{2}{7}a^{12}+\frac{3}{7}a^{11}-\frac{6}{7}a^{10}-\frac{2}{7}a^{9}+\frac{4}{7}a^{8}-\frac{1}{7}a^{7}-\frac{5}{7}a^{6}+\frac{3}{7}a^{5}+\frac{1}{7}a^{4}-\frac{2}{7}a^{3}-\frac{3}{7}a^{2}+\frac{6}{7}a+\frac{5}{7}$, $\frac{1}{2}a^{12}-a^{6}+1$, $\frac{1}{14}a^{22}-\frac{1}{7}a^{21}-\frac{3}{14}a^{20}-\frac{1}{14}a^{19}+\frac{1}{7}a^{18}+\frac{3}{14}a^{17}+\frac{1}{14}a^{16}-\frac{1}{7}a^{15}+\frac{2}{7}a^{14}+\frac{3}{7}a^{13}-\frac{5}{14}a^{12}-\frac{2}{7}a^{11}-\frac{3}{7}a^{10}-\frac{1}{7}a^{9}+\frac{2}{7}a^{8}-\frac{4}{7}a^{7}+\frac{1}{7}a^{6}+\frac{5}{7}a^{5}+\frac{4}{7}a^{4}+\frac{6}{7}a^{3}-\frac{5}{7}a^{2}-\frac{4}{7}a-\frac{1}{7}$, $\frac{1}{7}a^{22}+\frac{3}{14}a^{21}+\frac{1}{14}a^{20}-\frac{1}{7}a^{19}-\frac{3}{14}a^{18}+\frac{3}{7}a^{17}-\frac{5}{14}a^{16}+\frac{5}{7}a^{15}-\frac{3}{7}a^{14}-\frac{1}{7}a^{13}-\frac{5}{7}a^{12}+\frac{3}{7}a^{11}+\frac{1}{7}a^{10}-\frac{2}{7}a^{9}+\frac{4}{7}a^{8}-\frac{8}{7}a^{7}+\frac{2}{7}a^{6}+\frac{3}{7}a^{5}+\frac{8}{7}a^{4}-\frac{2}{7}a^{3}-\frac{3}{7}a^{2}-\frac{1}{7}a-\frac{9}{7}$, $\frac{4}{7}a^{22}-\frac{9}{14}a^{21}+\frac{11}{14}a^{20}-\frac{4}{7}a^{19}+\frac{9}{14}a^{18}-\frac{2}{7}a^{17}+\frac{1}{14}a^{16}+\frac{5}{14}a^{15}-\frac{5}{7}a^{14}+\frac{13}{14}a^{13}-\frac{5}{14}a^{12}+\frac{5}{7}a^{11}-\frac{3}{7}a^{10}+\frac{6}{7}a^{9}-\frac{5}{7}a^{8}+\frac{3}{7}a^{7}+\frac{1}{7}a^{6}-\frac{2}{7}a^{5}+\frac{11}{7}a^{4}-\frac{8}{7}a^{3}+\frac{9}{7}a^{2}-\frac{4}{7}a-\frac{15}{7}$, $\frac{2}{7}a^{22}-\frac{1}{14}a^{21}+\frac{1}{7}a^{20}+\frac{3}{14}a^{19}+\frac{1}{14}a^{18}-\frac{1}{7}a^{17}+\frac{2}{7}a^{16}-\frac{4}{7}a^{15}+\frac{9}{14}a^{14}-\frac{2}{7}a^{13}+\frac{4}{7}a^{12}-\frac{1}{7}a^{11}+\frac{2}{7}a^{10}-\frac{4}{7}a^{9}+\frac{1}{7}a^{8}-\frac{2}{7}a^{7}-\frac{3}{7}a^{6}+\frac{6}{7}a^{5}-\frac{5}{7}a^{4}+\frac{3}{7}a^{3}-\frac{6}{7}a^{2}-\frac{2}{7}a-\frac{11}{7}$, $\frac{5}{14}a^{22}-\frac{5}{7}a^{21}+\frac{3}{7}a^{20}-\frac{5}{14}a^{19}+\frac{3}{14}a^{18}+\frac{1}{14}a^{17}-\frac{1}{7}a^{16}-\frac{3}{14}a^{15}-\frac{1}{14}a^{14}-\frac{5}{14}a^{13}+\frac{5}{7}a^{12}-\frac{3}{7}a^{11}+\frac{6}{7}a^{10}-\frac{5}{7}a^{9}+\frac{3}{7}a^{8}+\frac{1}{7}a^{7}-\frac{2}{7}a^{6}+\frac{4}{7}a^{5}-\frac{1}{7}a^{4}+\frac{2}{7}a^{3}+\frac{10}{7}a^{2}-\frac{6}{7}a-\frac{5}{7}$ (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: | \( 1131394806.61 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{11}\cdot 1131394806.61 \cdot 1}{2\cdot\sqrt{1670424922585164788805622697952280576}}\cr\approx \mathstrut & 0.527447517157 \end{aligned}\] (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 | R | ${\href{/padicField/3.9.0.1}{9} }{,}\,{\href{/padicField/3.7.0.1}{7} }{,}\,{\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | $16{,}\,{\href{/padicField/5.5.0.1}{5} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | $17{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.11.0.1}{11} }{,}\,{\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.11.0.1}{11} }{,}\,{\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | $20{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.11.0.1}{11} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $23$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.7.0.1}{7} }^{2}{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/43.7.0.1}{7} }$ | ${\href{/padicField/47.9.0.1}{9} }{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | $19{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.11.0.1}{11} }{,}\,{\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.23.22.1 | $x^{23} + 2$ | $23$ | $1$ | $22$ | $C_{23}:C_{11}$ | $[\ ]_{23}^{11}$ |
\(8623\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
\(10045730659\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $21$ | $1$ | $21$ | $0$ | $C_{21}$ | $[\ ]^{21}$ | ||
\(4597557191821267\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $19$ | $1$ | $19$ | $0$ | $C_{19}$ | $[\ ]^{19}$ |