Normalized defining polynomial
\( x^{23} - x - 2 \)
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: | \(-87579030112206219473123894560276086784\) \(\medspace = -\,2^{23}\cdot 7\cdot 24177221\cdot 78687506363\cdot 783971054543\) | 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: | not computed | ||
Ramified primes: | \(2\), \(7\), \(24177221\), \(78687506363\), \(783971054543\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-20880\!\cdots\!97246}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$
Monogenic: | Yes | |
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: | $a^{12}+a+1$, $a^{21}+a^{19}+a^{17}+a^{15}+a^{13}+a^{11}+a^{9}+a^{7}+a^{6}+a^{5}+a^{4}+a^{3}+a^{2}+a+1$, $a^{22}-a^{21}+a^{20}-a^{19}+a^{18}-a^{17}+a^{16}-a^{15}+a^{14}-a^{13}+a^{12}-a^{11}+a^{10}-a^{9}+a^{8}-a^{7}+a^{6}-a^{5}+a^{4}-a^{3}+a^{2}-a-1$, $a^{22}-a^{20}+a^{18}-a^{16}+a^{14}-a^{12}+a^{10}-a^{8}+a^{7}-a^{5}+a^{3}-a-1$, $2a^{22}-2a^{21}+a^{20}-a^{18}+2a^{17}-2a^{16}+a^{15}-a^{13}+2a^{12}-2a^{11}+a^{10}+a^{9}-2a^{8}+2a^{7}-2a^{6}+a^{5}+a^{4}-2a^{3}+2a^{2}-2a-1$, $a^{19}-a^{11}-a^{7}-a^{6}+a^{5}-a^{3}-a^{2}+a+1$, $a^{22}+a^{21}+a^{17}+a^{16}-a^{13}-a^{9}-2a^{8}-a^{7}+a^{5}-a^{3}-a^{2}+1$, $a^{21}+a^{20}+a^{19}-a^{17}-a^{16}-a^{15}-a^{14}-a^{13}+a^{11}+2a^{10}+a^{9}+a^{8}-2a^{5}-3a^{4}-3a^{3}-a^{2}+1$, $a^{21}-a^{20}+a^{17}-a^{16}+a^{13}-a^{12}+a^{10}+a^{9}-a^{8}+2a^{6}-a^{4}+a^{3}+2a^{2}-a-1$, $2a^{22}-a^{21}-a^{20}+2a^{19}-2a^{18}+2a^{16}-3a^{15}+a^{14}+2a^{13}-3a^{12}+2a^{11}+a^{10}-3a^{9}+3a^{8}-3a^{6}+3a^{5}-a^{4}-2a^{3}+3a^{2}-2a-3$, $a^{21}-a^{19}-2a^{17}+2a^{13}+a^{12}+2a^{11}+a^{10}-2a^{7}-2a^{6}-2a^{5}-2a^{4}+2a+3$ (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: | \( 17404788077.7 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{11}\cdot 17404788077.7 \cdot 1}{2\cdot\sqrt{87579030112206219473123894560276086784}}\cr\approx \mathstrut & 1.12059124720 \end{aligned}\] (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 | ${\href{/padicField/3.9.0.1}{9} }{,}\,{\href{/padicField/3.7.0.1}{7} }{,}\,{\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | $16{,}\,{\href{/padicField/5.5.0.1}{5} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | R | $17{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.11.0.1}{11} }{,}\,{\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.11.0.1}{11} }{,}\,{\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.9.0.1}{9} }$ | $23$ | $17{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | $18{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.11.0.1}{11} }$ | $19{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
2.2.3.1 | $x^{2} + 4 x + 2$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
Deg $20$ | $2$ | $10$ | $20$ | ||||
\(7\) | $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
7.17.0.1 | $x^{17} + x + 4$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | |
\(24177221\) | $\Q_{24177221}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{24177221}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
\(78687506363\) | $\Q_{78687506363}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{78687506363}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | ||
\(783971054543\) | $\Q_{783971054543}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |