Normalized defining polynomial
\( x^{23} + 8x - 8 \)
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: | \(-99035448947556141209079331471444934656\) \(\medspace = -\,2^{22}\cdot 19\cdot 2617\cdot 310741\cdot 1528181322635255601073\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(44.87\) | 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: | $2^{22/23}19^{1/2}2617^{1/2}310741^{1/2}1528181322635255601073^{1/2}\approx 9429900711203802.0$ | ||
Ramified primes: | \(2\), \(19\), \(2617\), \(310741\), \(1528181322635255601073\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-23611\!\cdots\!99239}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}$, $\frac{1}{2}a^{9}$, $\frac{1}{2}a^{10}$, $\frac{1}{2}a^{11}$, $\frac{1}{2}a^{12}$, $\frac{1}{2}a^{13}$, $\frac{1}{2}a^{14}$, $\frac{1}{2}a^{15}$, $\frac{1}{4}a^{16}$, $\frac{1}{4}a^{17}$, $\frac{1}{4}a^{18}$, $\frac{1}{4}a^{19}$, $\frac{1}{4}a^{20}$, $\frac{1}{4}a^{21}$, $\frac{1}{4}a^{22}$
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)
| |
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: | $a-1$, $\frac{1}{2}a^{8}-a+1$, $\frac{1}{4}a^{22}+\frac{1}{4}a^{21}-\frac{1}{4}a^{19}-\frac{1}{4}a^{18}+\frac{1}{4}a^{16}+\frac{1}{2}a^{15}+\frac{1}{2}a^{14}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}+\frac{1}{2}a^{9}+\frac{1}{2}a^{8}-a+1$, $\frac{1}{4}a^{22}-\frac{1}{4}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-a^{13}-a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-a^{9}-a^{8}-a^{5}+a^{3}+3$, $\frac{1}{4}a^{22}+\frac{1}{4}a^{21}+\frac{1}{4}a^{20}-\frac{1}{4}a^{18}-\frac{1}{4}a^{17}-\frac{1}{4}a^{16}+\frac{1}{2}a^{13}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-a^{8}+a^{5}+a^{3}-a^{2}-a+1$, $\frac{1}{4}a^{21}+\frac{1}{2}a^{19}+\frac{1}{4}a^{17}+\frac{1}{2}a^{14}+\frac{1}{2}a^{12}-\frac{1}{2}a^{11}+\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-a^{6}+a^{5}-a^{4}+a^{3}-a^{2}-a+1$, $\frac{1}{4}a^{22}+\frac{1}{4}a^{21}+\frac{1}{4}a^{20}-\frac{1}{4}a^{17}+\frac{1}{2}a^{14}+\frac{1}{2}a^{13}+\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-a^{9}-a^{8}-a^{7}+a^{4}-a+1$, $\frac{1}{4}a^{22}+\frac{1}{4}a^{20}+\frac{1}{2}a^{19}+\frac{1}{2}a^{17}+\frac{1}{4}a^{16}+\frac{1}{2}a^{14}+\frac{1}{2}a^{13}-\frac{1}{2}a^{12}+a^{11}-\frac{1}{2}a^{9}+\frac{1}{2}a^{8}-a^{6}+a^{5}-a^{4}-a^{3}+2a^{2}-2a+1$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{19}+\frac{1}{2}a^{18}+\frac{3}{4}a^{17}-\frac{3}{4}a^{16}-\frac{1}{2}a^{15}+\frac{3}{2}a^{14}-\frac{3}{2}a^{12}+\frac{1}{2}a^{11}+\frac{3}{2}a^{10}-\frac{3}{2}a^{9}-\frac{1}{2}a^{8}+a^{7}-a^{5}+a^{4}-a^{3}+2a-1$, $\frac{3}{4}a^{22}+a^{21}+\frac{5}{4}a^{20}+\frac{3}{2}a^{19}+\frac{3}{2}a^{18}+\frac{5}{4}a^{17}+a^{16}+\frac{1}{2}a^{15}-\frac{1}{2}a^{13}-a^{12}-a^{11}-a^{10}-a^{9}-\frac{1}{2}a^{8}-a^{3}-a^{2}-2a+3$, $a^{22}+\frac{3}{4}a^{21}+\frac{3}{4}a^{20}+\frac{1}{2}a^{19}+\frac{1}{2}a^{18}+\frac{1}{4}a^{17}+\frac{1}{4}a^{16}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-a^{8}-a^{7}-2a^{6}-a^{5}-2a^{4}-a^{3}-2a^{2}+7$ (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: | \( 13784529892.7 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{11}\cdot 13784529892.7 \cdot 1}{2\cdot\sqrt{99035448947556141209079331471444934656}}\cr\approx \mathstrut & 0.834593838464 \end{aligned}\] (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 | ${\href{/padicField/3.9.0.1}{9} }{,}\,{\href{/padicField/3.7.0.1}{7} }{,}\,{\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | $15{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | $17{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | R | $22{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $23$ | $22{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.14.0.1}{14} }{,}\,{\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | $17{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.9.0.1}{9} }{,}\,{\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | $17{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.14.0.1}{14} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 | $x^{23} + 2$ | $23$ | $1$ | $22$ | $C_{23}:C_{11}$ | $[\ ]_{23}^{11}$ |
\(19\) | $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
19.3.0.1 | $x^{3} + 4 x + 17$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
19.17.0.1 | $x^{17} + 2 x + 17$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | |
\(2617\) | $\Q_{2617}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{2617}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | ||
\(310741\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | ||
\(152\!\cdots\!073\) | $\Q_{15\!\cdots\!73}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ |