Normalized defining polynomial
\( x^{22} - 12 x^{20} + 64 x^{18} - 200 x^{16} + 403 x^{14} - 544 x^{12} + 510 x^{10} - 358 x^{8} + 197 x^{6} - 73 x^{4} + 14 x^{2} - 1 \)
Invariants
Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[14, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(13471519681654627790892423970816\) \(\medspace = 2^{22}\cdot 1792166448977^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(26.00\) | 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: | \(2\), \(1792166448977\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $17$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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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)
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Fundamental units: | $a$, $a^{20}-11a^{18}+53a^{16}-147a^{14}+256a^{12}-288a^{10}+222a^{8}-136a^{6}+61a^{4}-12a^{2}+2$, $4a^{21}-49a^{19}+264a^{17}-825a^{15}+1644a^{13}-2160a^{11}+1930a^{9}-1281a^{7}+669a^{5}-212a^{3}+23a$, $a^{20}-11a^{18}+54a^{16}-156a^{14}+291a^{12}-365a^{10}+324a^{8}-220a^{6}+115a^{4}-40a^{2}+6$, $4a^{20}-46a^{18}+234a^{16}-693a^{14}+1309a^{12}-1630a^{10}+1393a^{8}-902a^{6}+455a^{4}-133a^{2}+14$, $a^{3}-2a$, $6a^{20}-71a^{18}+371a^{16}-1128a^{14}+2191a^{12}-2813a^{10}+2476a^{8}-1637a^{6}+845a^{4}-264a^{2}+31$, $3a^{20}-38a^{18}+210a^{16}-669a^{14}+1353a^{12}-1795a^{10}+1606a^{8}-1061a^{6}+555a^{4}-174a^{2}+18$, $a^{21}+a^{20}-11a^{19}-11a^{18}+53a^{17}+53a^{16}-147a^{15}-147a^{14}+256a^{13}+256a^{12}-288a^{11}-288a^{10}+222a^{9}+222a^{8}-136a^{7}-136a^{6}+61a^{5}+61a^{4}-12a^{3}-12a^{2}+2a+2$, $8a^{21}-4a^{20}-94a^{19}+48a^{18}+489a^{17}-253a^{16}-1482a^{15}+772a^{14}+2869a^{13}-1497a^{12}-3666a^{11}+1904a^{10}+3199a^{9}-1642a^{8}-2086a^{7}+1059a^{6}+1065a^{5}-534a^{4}-324a^{3}+154a^{2}+33a-14$, $a^{2}-a-1$, $a^{21}+5a^{20}-12a^{19}-60a^{18}+64a^{17}+317a^{16}-200a^{15}-972a^{14}+403a^{13}+1900a^{12}-544a^{11}-2448a^{10}+510a^{9}+2152a^{8}-358a^{7}-1417a^{6}+197a^{5}+730a^{4}-73a^{3}-223a^{2}+13a+24$, $6a^{21}+a^{20}-72a^{19}-11a^{18}+381a^{17}+54a^{16}-1172a^{15}-156a^{14}+2303a^{13}+291a^{12}-2992a^{11}-365a^{10}+2662a^{9}+324a^{8}-1775a^{7}-220a^{6}+927a^{5}+115a^{4}-296a^{3}-40a^{2}+37a+6$, $5a^{21}-4a^{20}-55a^{19}+46a^{18}+269a^{17}-234a^{16}-769a^{15}+693a^{14}+1405a^{13}-1309a^{12}-1699a^{11}+1630a^{10}+1428a^{9}-1393a^{8}-921a^{7}+902a^{6}+456a^{5}-455a^{4}-129a^{3}+132a^{2}+15a-13$, $a^{20}-11a^{18}+54a^{16}-156a^{14}+291a^{12}-365a^{10}+324a^{8}-220a^{6}+115a^{4}-40a^{2}-a+6$, $2a^{21}-25a^{19}+137a^{17}-435a^{15}+882a^{13}-1183a^{11}+1083a^{9}-735a^{7}+390a^{5}-131a^{3}+17a+1$, $8a^{21}-a^{20}-93a^{19}+11a^{18}+479a^{17}-53a^{16}-1438a^{15}+147a^{14}+2758a^{13}-256a^{12}-3494a^{11}+288a^{10}+3034a^{9}-222a^{8}-1982a^{7}+136a^{6}+1011a^{5}-61a^{4}-304a^{3}+12a^{2}+34a-2$ (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)]
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Regulator: | \( 45766346.4516 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{4}\cdot 45766346.4516 \cdot 1}{2\cdot\sqrt{13471519681654627790892423970816}}\cr\approx \mathstrut & 0.159201577209 \end{aligned}\] (assuming GRH)
Galois group
$C_2^{10}.S_{11}$ (as 22T51):
A non-solvable group of order 40874803200 |
The 376 conjugacy class representatives for $C_2^{10}.S_{11}$ are not computed |
Character table for $C_2^{10}.S_{11}$ is not computed |
Intermediate fields
11.7.1792166448977.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 22 sibling: | data not computed |
Degree 44 sibling: | data not computed |
Minimal sibling: | 22.14.2569340566868698646254256448691034157941557089947947038410673463363632924132693018179259270357029410958376322699590917878586229919830027272192.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.5.0.1}{5} }^{2}$ | ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.10.0.1}{10} }$ | ${\href{/padicField/7.7.0.1}{7} }^{2}{,}\,{\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.11.0.1}{11} }^{2}$ | ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.10.0.1}{10} }$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.10.0.1}{10} }$ | ${\href{/padicField/19.8.0.1}{8} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | $20{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.10.0.1}{10} }$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.11.0.1}{11} }^{2}$ | ${\href{/padicField/47.10.0.1}{10} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.14.0.1}{14} }{,}\,{\href{/padicField/59.8.0.1}{8} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $22$ | $2$ | $11$ | $22$ | |||
\(1792166448977\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ |