Normalized defining polynomial
\( x^{23} - x - 4 \)
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: | \(-87579030453634015434594065117161697047\) \(\medspace = -\,7901\cdot 159683\cdot 69415968602603755097356934209\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(44.63\) | 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: | $7901^{1/2}159683^{1/2}69415968602603755097356934209^{1/2}\approx 9.358366868937871e+18$ | ||
Ramified primes: | \(7901\), \(159683\), \(69415968602603755097356934209\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-87579\!\cdots\!97047}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{7}$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{8}$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{9}$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{10}$, $\frac{1}{2}a^{22}-\frac{1}{2}a^{11}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
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: | $\frac{1}{2}a^{12}-\frac{1}{2}a-1$, $\frac{1}{2}a^{22}-\frac{1}{2}a^{21}+\frac{1}{2}a^{20}-\frac{1}{2}a^{19}+\frac{1}{2}a^{18}-\frac{1}{2}a^{17}+\frac{1}{2}a^{16}-\frac{1}{2}a^{15}+\frac{1}{2}a^{14}-\frac{1}{2}a^{13}+\frac{1}{2}a^{12}-\frac{1}{2}a^{11}+\frac{1}{2}a^{10}-\frac{1}{2}a^{9}+\frac{1}{2}a^{8}-\frac{1}{2}a^{7}+\frac{1}{2}a^{6}-\frac{1}{2}a^{5}+\frac{1}{2}a^{4}-\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{1}{2}a-1$, $\frac{1}{2}a^{16}+\frac{1}{2}a^{14}+\frac{1}{2}a^{13}-a^{8}-\frac{1}{2}a^{5}+\frac{1}{2}a^{3}+\frac{1}{2}a^{2}+1$, $\frac{1}{2}a^{22}-\frac{1}{2}a^{21}-\frac{1}{2}a^{20}+\frac{1}{2}a^{19}+\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}+a^{15}-a^{13}+\frac{3}{2}a^{11}-\frac{1}{2}a^{10}-\frac{3}{2}a^{9}+\frac{1}{2}a^{8}+\frac{3}{2}a^{7}-\frac{1}{2}a^{6}-\frac{3}{2}a^{5}+a^{4}+a^{3}-a^{2}-a+1$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{19}+\frac{1}{2}a^{17}-\frac{1}{2}a^{15}-\frac{1}{2}a^{12}+\frac{1}{2}a^{10}-\frac{1}{2}a^{8}+\frac{1}{2}a^{6}-\frac{1}{2}a^{4}+a^{2}+\frac{1}{2}a-1$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{18}+\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}+a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}+a-1$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}+\frac{1}{2}a^{16}+\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}+\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}+\frac{3}{2}a^{6}+\frac{1}{2}a^{5}-\frac{5}{2}a^{4}+\frac{1}{2}a^{3}+\frac{3}{2}a^{2}-\frac{1}{2}a-1$, $\frac{1}{2}a^{19}+\frac{1}{2}a^{17}-\frac{1}{2}a^{16}+\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{12}+\frac{1}{2}a^{8}+\frac{1}{2}a^{6}+\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-a^{2}+\frac{1}{2}a-1$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{19}+\frac{1}{2}a^{17}-\frac{1}{2}a^{15}+a^{13}-a^{11}+\frac{3}{2}a^{9}-\frac{1}{2}a^{8}-a^{7}+\frac{1}{2}a^{6}+a^{5}+\frac{1}{2}a^{4}-2a^{3}+a^{2}+a-1$, $\frac{1}{2}a^{22}+\frac{1}{2}a^{21}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}+a^{15}+\frac{1}{2}a^{14}+a^{13}-\frac{1}{2}a^{12}-\frac{3}{2}a^{11}-\frac{3}{2}a^{10}-a^{9}+\frac{5}{2}a^{7}+\frac{3}{2}a^{6}+2a^{5}-\frac{5}{2}a^{3}-2a^{2}-\frac{3}{2}a-1$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{20}+\frac{1}{2}a^{18}+\frac{1}{2}a^{17}+\frac{1}{2}a^{15}+\frac{1}{2}a^{12}-\frac{1}{2}a^{10}+\frac{1}{2}a^{9}+a^{8}-\frac{1}{2}a^{7}+\frac{1}{2}a^{6}+a^{5}+\frac{1}{2}a^{4}+2a^{2}-\frac{1}{2}a-1$ (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: | \( 10664511959.7 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{11}\cdot 10664511959.7 \cdot 1}{2\cdot\sqrt{87579030453634015434594065117161697047}}\cr\approx \mathstrut & 0.686624777102 \end{aligned}\] (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 | $20{,}\,{\href{/padicField/2.2.0.1}{2} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | ${\href{/padicField/3.9.0.1}{9} }{,}\,{\href{/padicField/3.7.0.1}{7} }{,}\,{\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/7.3.0.1}{3} }$ | ${\href{/padicField/11.7.0.1}{7} }^{2}{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | $23$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | $17{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | $23$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.11.0.1}{11} }{,}\,{\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.9.0.1}{9} }{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | $20{,}\,{\href{/padicField/53.3.0.1}{3} }$ | $15{,}\,{\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(7901\) | $\Q_{7901}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
\(159683\) | $\Q_{159683}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{159683}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{159683}$ | $x$ | $1$ | $1$ | $0$ | 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 $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
\(694\!\cdots\!209\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
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 $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |