Normalized defining polynomial
\( x^{23} - 9x - 4 \)
Invariants
Degree: | $23$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[3, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(721467443006110925268556409136101552737959841\) \(\medspace = 29357399\cdot 2802945917\cdot 8767675026551455678892251027\) | 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: | \(89.20\) | 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: | $29357399^{1/2}2802945917^{1/2}8767675026551455678892251027^{1/2}\approx 2.6860145997483168e+22$ | ||
Ramified primes: | \(29357399\), \(2802945917\), \(8767675026551455678892251027\) | 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: | $\Q(\sqrt{72146\!\cdots\!59841}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{7}$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{8}$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{9}$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{10}$, $\frac{1}{2}a^{22}-\frac{1}{2}a^{11}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $12$ | 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: | $\frac{3}{2}a^{12}-\frac{9}{2}a-1$, $\frac{1}{2}a^{21}+\frac{1}{2}a^{20}+\frac{1}{2}a^{19}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-a^{16}+\frac{1}{2}a^{14}+\frac{3}{2}a^{13}+a^{12}-\frac{3}{2}a^{10}-\frac{3}{2}a^{9}-\frac{3}{2}a^{8}+\frac{3}{2}a^{7}+\frac{5}{2}a^{6}+4a^{5}+a^{4}-\frac{3}{2}a^{3}-\frac{9}{2}a^{2}-4a-1$, $\frac{1}{2}a^{22}-a^{21}-a^{18}-\frac{1}{2}a^{17}+\frac{1}{2}a^{16}-a^{15}-a^{14}+\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{3}{2}a^{11}+a^{10}-a^{9}-2a^{8}+a^{7}-\frac{1}{2}a^{6}-\frac{7}{2}a^{5}-a^{4}+a^{3}-\frac{11}{2}a^{2}-\frac{5}{2}a-5$, $4a^{22}-2a^{21}+a^{20}-a^{19}+\frac{1}{2}a^{17}-2a^{16}+\frac{3}{2}a^{15}+\frac{1}{2}a^{14}+a^{13}+a^{12}-a^{11}+2a^{10}-4a^{9}-2a^{8}+a^{7}-\frac{3}{2}a^{6}+3a^{5}-\frac{3}{2}a^{4}+\frac{11}{2}a^{3}+a^{2}-5a-31$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{20}-\frac{1}{2}a^{19}+\frac{3}{2}a^{18}-a^{17}-a^{16}+\frac{5}{2}a^{15}-2a^{14}-\frac{1}{2}a^{13}+\frac{5}{2}a^{12}-2a^{11}-\frac{1}{2}a^{10}+\frac{5}{2}a^{9}-\frac{5}{2}a^{8}+\frac{1}{2}a^{7}+2a^{6}-2a^{5}+\frac{1}{2}a^{4}+\frac{3}{2}a^{2}-\frac{3}{2}a-1$, $\frac{11}{2}a^{22}-2a^{21}+\frac{3}{2}a^{20}+\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}+\frac{1}{2}a^{9}+a^{8}+\frac{5}{2}a^{7}+\frac{3}{2}a^{6}+a^{5}-\frac{3}{2}a^{4}-\frac{7}{2}a^{3}-4a^{2}-\frac{7}{2}a-49$, $\frac{5}{2}a^{22}-a^{21}-\frac{5}{2}a^{20}+3a^{19}+a^{18}-\frac{9}{2}a^{17}+3a^{16}+\frac{7}{2}a^{15}-7a^{14}+\frac{3}{2}a^{13}+6a^{12}-\frac{13}{2}a^{11}-a^{10}+\frac{17}{2}a^{9}-5a^{8}-6a^{7}+\frac{21}{2}a^{6}-4a^{5}-\frac{19}{2}a^{4}+16a^{3}-\frac{5}{2}a^{2}-15a-5$, $\frac{1}{2}a^{22}-a^{21}+a^{20}-a^{19}+a^{17}-2a^{16}+2a^{15}-2a^{14}-\frac{1}{2}a^{13}+\frac{3}{2}a^{12}-\frac{7}{2}a^{11}+3a^{10}-2a^{9}-a^{8}+2a^{7}-3a^{6}+3a^{5}-3a^{4}-\frac{3}{2}a^{2}-\frac{7}{2}a-1$, $2a^{22}+a^{21}+\frac{3}{2}a^{20}+a^{19}+\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{3}{2}a^{16}-2a^{15}-2a^{14}-2a^{13}-\frac{3}{2}a^{12}+a^{10}+\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{5}{2}a^{6}-\frac{9}{2}a^{5}-4a^{4}-2a^{3}+a^{2}+\frac{13}{2}a-5$, $\frac{3}{2}a^{22}-\frac{3}{2}a^{21}+2a^{20}-3a^{19}+2a^{18}-\frac{1}{2}a^{16}+\frac{3}{2}a^{15}-4a^{14}+5a^{13}-\frac{7}{2}a^{12}+\frac{5}{2}a^{11}-\frac{7}{2}a^{10}+2a^{9}+2a^{8}-a^{7}-a^{6}-\frac{5}{2}a^{5}+\frac{7}{2}a^{4}+a^{2}-\frac{15}{2}a-3$, $\frac{15}{2}a^{22}-\frac{9}{2}a^{21}-\frac{3}{2}a^{20}+7a^{19}-\frac{23}{2}a^{18}+\frac{25}{2}a^{17}-9a^{16}+\frac{3}{2}a^{15}+8a^{14}-\frac{31}{2}a^{13}+20a^{12}-\frac{33}{2}a^{11}+\frac{15}{2}a^{10}+\frac{13}{2}a^{9}-22a^{8}+\frac{57}{2}a^{7}-\frac{61}{2}a^{6}+18a^{5}+\frac{7}{2}a^{4}-24a^{3}+\frac{91}{2}a^{2}-46a-31$, $\frac{3}{2}a^{22}+7a^{21}+12a^{20}+14a^{19}+\frac{23}{2}a^{18}+4a^{17}-6a^{16}-\frac{33}{2}a^{15}-\frac{49}{2}a^{14}-24a^{13}-\frac{31}{2}a^{12}+\frac{1}{2}a^{11}+21a^{10}+39a^{9}+46a^{8}+\frac{77}{2}a^{7}+16a^{6}-21a^{5}-\frac{119}{2}a^{4}-\frac{171}{2}a^{3}-90a^{2}-\frac{127}{2}a-17$ (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: | \( 52936041498100 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{10}\cdot 52936041498100 \cdot 1}{2\cdot\sqrt{721467443006110925268556409136101552737959841}}\cr\approx \mathstrut & 0.755965137962164 \end{aligned}\] (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 | $20{,}\,{\href{/padicField/2.2.0.1}{2} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | ${\href{/padicField/3.11.0.1}{11} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/5.5.0.1}{5} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.11.0.1}{11} }{,}\,{\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ | $21{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.9.0.1}{9} }$ | ${\href{/padicField/23.11.0.1}{11} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.13.0.1}{13} }{,}\,{\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.11.0.1}{11} }{,}\,{\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | $18{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.14.0.1}{14} }{,}\,{\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.9.0.1}{9} }{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | $22{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(29357399\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | ||
\(2802945917\) | $\Q_{2802945917}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{2802945917}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $17$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | ||
\(876\!\cdots\!027\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $19$ | $1$ | $19$ | $0$ | $C_{19}$ | $[\ ]^{19}$ |