Normalized defining polynomial
\( x^{23} + 3x - 6 \)
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: |
\(-2748322807307023618750300446349325480882161582080\)
\(\medspace = -\,2^{23}\cdot 3^{22}\cdot 5\cdot 67\cdot 7457\cdot 701609\cdot 5956705599335914343\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(127.66\) | 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: |
\(2\), \(3\), \(5\), \(67\), \(7457\), \(701609\), \(5956705599335914343\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-20880\!\cdots\!50530}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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: |
$a^{22}-2a^{21}+3a^{20}-4a^{19}+4a^{18}-3a^{17}+a^{16}+a^{15}-3a^{14}+5a^{13}-7a^{12}+8a^{11}-7a^{10}+4a^{9}-4a^{7}+8a^{6}-12a^{5}+15a^{4}-15a^{3}+11a^{2}-4a-1$, $3a^{22}+19a^{21}-38a^{20}+40a^{19}-23a^{18}-13a^{17}+45a^{16}-65a^{15}+54a^{14}-13a^{13}-44a^{12}+85a^{11}-98a^{10}+56a^{9}+21a^{8}-104a^{7}+148a^{6}-132a^{5}+34a^{4}+95a^{3}-203a^{2}+231a-139$, $a^{21}+2a^{20}+5a^{19}+8a^{18}+9a^{17}+8a^{16}+7a^{15}+7a^{14}+9a^{13}+9a^{12}+5a^{11}-2a^{10}-9a^{9}-10a^{8}-6a^{7}-3a^{6}-7a^{5}-13a^{4}-24a^{3}-23a^{2}-15a-5$, $13a^{21}-31a^{20}+40a^{19}-48a^{18}+45a^{17}+a^{16}+49a^{15}-89a^{14}+58a^{13}-21a^{12}+26a^{11}-6a^{10}+88a^{9}-93a^{8}-32a^{7}+88a^{6}-4a^{5}+10a^{4}+14a^{3}+77a^{2}-253a+223$, $31a^{22}+14a^{21}-8a^{20}-31a^{19}-48a^{18}-57a^{17}-57a^{16}-52a^{15}-37a^{14}-11a^{13}+19a^{12}+52a^{11}+72a^{10}+67a^{9}+26a^{8}-37a^{7}-106a^{6}-161a^{5}-166a^{4}-123a^{3}-42a^{2}+47a+215$, $28a^{22}+30a^{21}+26a^{20}+19a^{19}+21a^{18}-14a^{17}-21a^{16}-24a^{15}-27a^{14}-39a^{13}+2a^{12}+12a^{11}+11a^{10}+31a^{9}+46a^{8}+2a^{7}-21a^{6}-5a^{5}-65a^{4}-82a^{3}-48a^{2}-2a+35$, $5a^{22}-27a^{21}+81a^{20}-99a^{19}+132a^{18}-114a^{17}+102a^{16}-36a^{15}-20a^{14}+119a^{13}-174a^{12}+251a^{11}-251a^{10}+243a^{9}-143a^{8}+35a^{7}+138a^{6}-294a^{5}+449a^{4}-516a^{3}+537a^{2}-408a+241$, $59a^{22}-29a^{21}-21a^{20}+78a^{19}-99a^{18}+80a^{17}-a^{16}-78a^{15}+133a^{14}-151a^{13}+69a^{12}+39a^{11}-185a^{10}+230a^{9}-182a^{8}+23a^{7}+156a^{6}-331a^{5}+347a^{4}-197a^{3}-105a^{2}+409a-349$, $3689a^{22}-3079a^{21}-4055a^{20}+4073a^{19}+4380a^{18}-5259a^{17}-4647a^{16}+6712a^{15}+4816a^{14}-8434a^{13}-4826a^{12}+10475a^{11}+4662a^{10}-12893a^{9}-4162a^{8}+15661a^{7}+3374a^{6}-18926a^{5}-1993a^{4}+22562a^{3}+119a^{2}-26777a+13747$, $12a^{22}-11a^{21}+17a^{20}-15a^{19}+6a^{18}-5a^{17}-16a^{16}+14a^{15}-36a^{14}+17a^{13}-36a^{12}+14a^{11}-20a^{10}-13a^{9}+17a^{8}-49a^{7}+68a^{6}-65a^{5}+57a^{4}-24a^{3}+11a^{2}+51a-19$, $8a^{22}-7a^{21}+5a^{20}+22a^{19}+32a^{18}+56a^{17}+96a^{16}+120a^{15}+103a^{14}+88a^{13}+86a^{12}+52a^{11}-2a^{10}-19a^{9}+15a^{8}+39a^{7}+43a^{6}+109a^{5}+202a^{4}+213a^{3}+175a^{2}+162a+193$
| 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: | \( 4214765620360000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{11}\cdot 4214765620360000 \cdot 1}{2\cdot\sqrt{2748322807307023618750300446349325480882161582080}}\cr\approx \mathstrut & 1.53185659217590 \end{aligned}\] (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 | R | R | R | $20{,}\,{\href{/padicField/7.3.0.1}{3} }$ | $20{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | $15{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.13.0.1}{13} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.13.0.1}{13} }{,}\,{\href{/padicField/31.10.0.1}{10} }$ | $20{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.14.0.1}{14} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | $20{,}\,{\href{/padicField/47.3.0.1}{3} }$ | $22{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.11.0.1}{11} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 | $[\ ]$ |
2.2.3.4 | $x^{2} + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
Deg $20$ | $2$ | $10$ | $20$ | ||||
\(3\)
| 3.23.22.1 | $x^{23} + 3$ | $23$ | $1$ | $22$ | $C_{23}:C_{11}$ | $[\ ]_{23}^{11}$ |
\(5\)
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.8.0.1 | $x^{8} + x^{4} + 3 x^{2} + 4 x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
5.12.0.1 | $x^{12} + x^{7} + x^{6} + 4 x^{4} + 4 x^{3} + 3 x^{2} + 2 x + 2$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
\(67\)
| $\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
67.2.1.1 | $x^{2} + 134$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
67.20.0.1 | $x^{20} - 2 x + 7$ | $1$ | $20$ | $0$ | 20T1 | $[\ ]^{20}$ | |
\(7457\)
| $\Q_{7457}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
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 $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | ||
\(701609\)
| $\Q_{701609}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
\(5956705599335914343\)
| $\Q_{5956705599335914343}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ |