Normalized defining polynomial
\( x^{23} - 2 x^{22} - x^{21} + 5 x^{20} - x^{19} - 4 x^{18} + 6 x^{15} + 6 x^{14} - 17 x^{13} - 10 x^{12} + 31 x^{11} + 5 x^{10} - 32 x^{9} + 25 x^{7} - 17 x^{5} + 2 x^{4} + 9 x^{3} - 3 x^{2} - 2 x + 1 \)
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: | \(-459177995857290463522143518056811108\) \(\medspace = -\,2^{2}\cdot 1021\cdot 425123\cdot 264472629367076822658099919\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(35.52\) | 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: | \(2\), \(1021\), \(425123\), \(264472629367076822658099919\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-11479\!\cdots\!02777}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{11}a^{22}+\frac{4}{11}a^{21}+\frac{1}{11}a^{20}-\frac{1}{11}a^{18}+\frac{1}{11}a^{17}-\frac{5}{11}a^{16}+\frac{3}{11}a^{15}+\frac{2}{11}a^{14}-\frac{4}{11}a^{13}+\frac{3}{11}a^{12}-\frac{3}{11}a^{11}+\frac{2}{11}a^{10}-\frac{5}{11}a^{9}+\frac{4}{11}a^{8}+\frac{2}{11}a^{7}+\frac{4}{11}a^{6}+\frac{2}{11}a^{5}-\frac{5}{11}a^{4}+\frac{5}{11}a^{3}-\frac{5}{11}a^{2}-\frac{2}{11}$
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: | $a$, $a^{22}-3a^{21}+a^{20}+5a^{19}-4a^{18}-2a^{17}+8a^{14}+2a^{13}-21a^{12}+a^{11}+33a^{10}-13a^{9}-25a^{8}+11a^{7}+17a^{6}-8a^{5}-11a^{4}+8a^{3}+4a^{2}-4a+1$, $\frac{90}{11}a^{22}-\frac{102}{11}a^{21}-\frac{185}{11}a^{20}+26a^{19}+\frac{185}{11}a^{18}-\frac{196}{11}a^{17}-\frac{219}{11}a^{16}-\frac{203}{11}a^{15}+\frac{400}{11}a^{14}+\frac{938}{11}a^{13}-\frac{709}{11}a^{12}-\frac{1634}{11}a^{11}+\frac{1291}{11}a^{10}+\frac{1761}{11}a^{9}-\frac{1224}{11}a^{8}-\frac{1316}{11}a^{7}+\frac{976}{11}a^{6}+\frac{1049}{11}a^{5}-\frac{527}{11}a^{4}-\frac{430}{11}a^{3}+\frac{364}{11}a^{2}+11a-\frac{59}{11}$, $6a^{22}-16a^{21}+5a^{20}+26a^{19}-24a^{18}-6a^{17}+4a^{16}-4a^{15}+38a^{14}+10a^{13}-106a^{12}+14a^{11}+171a^{10}-90a^{9}-122a^{8}+86a^{7}+83a^{6}-58a^{5}-58a^{4}+53a^{3}+14a^{2}-27a+8$, $\frac{18}{11}a^{22}-\frac{27}{11}a^{21}-\frac{48}{11}a^{20}+9a^{19}+\frac{37}{11}a^{18}-\frac{103}{11}a^{17}-\frac{46}{11}a^{16}-\frac{12}{11}a^{15}+\frac{146}{11}a^{14}+\frac{203}{11}a^{13}-\frac{298}{11}a^{12}-\frac{439}{11}a^{11}+\frac{531}{11}a^{10}+\frac{504}{11}a^{9}-\frac{599}{11}a^{8}-\frac{382}{11}a^{7}+\frac{402}{11}a^{6}+\frac{333}{11}a^{5}-\frac{266}{11}a^{4}-\frac{174}{11}a^{3}+\frac{130}{11}a^{2}+2a-\frac{14}{11}$, $\frac{34}{11}a^{22}-\frac{40}{11}a^{21}-\frac{65}{11}a^{20}+10a^{19}+\frac{54}{11}a^{18}-\frac{76}{11}a^{17}-\frac{60}{11}a^{16}-\frac{63}{11}a^{15}+\frac{134}{11}a^{14}+\frac{315}{11}a^{13}-\frac{283}{11}a^{12}-\frac{542}{11}a^{11}+\frac{552}{11}a^{10}+\frac{556}{11}a^{9}-\frac{557}{11}a^{8}-\frac{394}{11}a^{7}+\frac{466}{11}a^{6}+\frac{321}{11}a^{5}-\frac{280}{11}a^{4}-\frac{127}{11}a^{3}+\frac{171}{11}a^{2}+3a-\frac{46}{11}$, $\frac{17}{11}a^{22}-\frac{20}{11}a^{21}-\frac{38}{11}a^{20}+4a^{19}+\frac{60}{11}a^{18}-\frac{27}{11}a^{17}-\frac{74}{11}a^{16}-\frac{59}{11}a^{15}+\frac{67}{11}a^{14}+\frac{229}{11}a^{13}-\frac{81}{11}a^{12}-\frac{370}{11}a^{11}+\frac{122}{11}a^{10}+\frac{399}{11}a^{9}-\frac{108}{11}a^{8}-\frac{340}{11}a^{7}+\frac{112}{11}a^{6}+\frac{210}{11}a^{5}-\frac{41}{11}a^{4}-\frac{124}{11}a^{3}+\frac{36}{11}a^{2}+4a-\frac{23}{11}$, $\frac{73}{11}a^{22}-\frac{93}{11}a^{21}-\frac{180}{11}a^{20}+27a^{19}+\frac{202}{11}a^{18}-\frac{290}{11}a^{17}-\frac{255}{11}a^{16}-\frac{89}{11}a^{15}+\frac{476}{11}a^{14}+\frac{852}{11}a^{13}-\frac{859}{11}a^{12}-\frac{1726}{11}a^{11}+\frac{1466}{11}a^{10}+\frac{2077}{11}a^{9}-\frac{1556}{11}a^{8}-\frac{1768}{11}a^{7}+\frac{1161}{11}a^{6}+\frac{1367}{11}a^{5}-\frac{662}{11}a^{4}-\frac{735}{11}a^{3}+\frac{350}{11}a^{2}+18a-\frac{91}{11}$, $\frac{80}{11}a^{22}-\frac{109}{11}a^{21}-\frac{129}{11}a^{20}+24a^{19}+\frac{118}{11}a^{18}-\frac{206}{11}a^{17}-\frac{169}{11}a^{16}-\frac{90}{11}a^{15}+\frac{369}{11}a^{14}+\frac{714}{11}a^{13}-\frac{805}{11}a^{12}-\frac{1252}{11}a^{11}+\frac{1425}{11}a^{10}+\frac{1327}{11}a^{9}-\frac{1407}{11}a^{8}-\frac{1061}{11}a^{7}+\frac{1255}{11}a^{6}+\frac{776}{11}a^{5}-\frac{708}{11}a^{4}-\frac{381}{11}a^{3}+\frac{447}{11}a^{2}+8a-\frac{127}{11}$, $4a^{22}-4a^{21}-12a^{20}+13a^{19}+18a^{18}-13a^{17}-20a^{16}-11a^{15}+24a^{14}+59a^{13}-31a^{12}-115a^{11}+46a^{10}+145a^{9}-47a^{8}-127a^{7}+29a^{6}+89a^{5}-11a^{4}-52a^{3}+6a^{2}+13a-2$, $\frac{59}{11}a^{22}-\frac{94}{11}a^{21}-\frac{106}{11}a^{20}+24a^{19}+\frac{73}{11}a^{18}-\frac{260}{11}a^{17}-\frac{108}{11}a^{16}+\frac{1}{11}a^{15}+\frac{349}{11}a^{14}+\frac{512}{11}a^{13}-\frac{835}{11}a^{12}-\frac{1035}{11}a^{11}+\frac{1548}{11}a^{10}+\frac{1102}{11}a^{9}-\frac{1733}{11}a^{8}-\frac{817}{11}a^{7}+\frac{1446}{11}a^{6}+\frac{580}{11}a^{5}-\frac{955}{11}a^{4}-\frac{255}{11}a^{3}+\frac{486}{11}a^{2}+a-\frac{140}{11}$ (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: | \( 583250036.6 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{11}\cdot 583250036.6 \cdot 1}{2\cdot\sqrt{459177995857290463522143518056811108}}\cr\approx \mathstrut & 0.5186122759 \end{aligned}\] (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 | $17{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | $21{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.14.0.1}{14} }{,}\,{\href{/padicField/7.9.0.1}{9} }$ | $17{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.11.0.1}{11} }{,}\,{\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.14.0.1}{14} }{,}\,{\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.9.0.1}{9} }$ | ${\href{/padicField/43.9.0.1}{9} }{,}\,{\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.11.0.1}{11} }{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | $15{,}\,{\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
2.21.0.1 | $x^{21} + x^{6} + x^{5} + x^{2} + 1$ | $1$ | $21$ | $0$ | $C_{21}$ | $[\ ]^{21}$ | |
\(1021\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | ||
\(425123\) | $\Q_{425123}$ | $x$ | $1$ | $1$ | $0$ | 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 $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
\(264\!\cdots\!919\) | $\Q_{26\!\cdots\!19}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $17$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ |