Normalized defining polynomial
\( x^{22} - 15 x^{20} + 91 x^{18} - 279 x^{16} + 417 x^{14} - 130 x^{12} - 453 x^{10} + 544 x^{8} - 66 x^{6} - 185 x^{4} + 75 x^{2} - 1 \)
Invariants
Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[18, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(39034415726392643532837005585022976\) \(\medspace = 2^{22}\cdot 137^{2}\cdot 293^{2}\cdot 11093^{2}\cdot 216649^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(37.35\) | 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: | \(2\), \(137\), \(293\), \(11093\), \(216649\) | 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)
| |
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: | $19$ | 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)
| |
Fundamental units: | $3a^{21}-40a^{19}+206a^{17}-489a^{15}+411a^{13}+356a^{11}-817a^{9}+222a^{7}+278a^{5}-118a^{3}-6a$, $a^{21}-14a^{19}+77a^{17}-202a^{15}+215a^{13}+85a^{11}-369a^{9}+182a^{7}+102a^{5}-80a^{3}+7a$, $a^{2}-1$, $3a^{21}-41a^{19}+218a^{17}-542a^{15}+507a^{13}+332a^{11}-941a^{9}+315a^{7}+313a^{5}-143a^{3}-10a$, $4a^{20}-55a^{18}+295a^{16}-744a^{14}+722a^{12}+418a^{10}-1317a^{8}+511a^{6}+412a^{4}-235a^{2}+3$, $a^{21}-15a^{19}+89a^{17}-255a^{15}+311a^{13}+61a^{11}-493a^{9}+275a^{7}+137a^{5}-106a^{3}+6a$, $7a^{20}-95a^{18}+501a^{16}-1234a^{14}+1142a^{12}+748a^{10}-2117a^{8}+767a^{6}+654a^{4}-364a^{2}+7$, $6a^{21}-81a^{19}+425a^{17}-1043a^{15}+970a^{13}+601a^{11}-1761a^{9}+686a^{7}+512a^{5}-320a^{3}+13a$, $a^{21}-14a^{19}+77a^{17}-202a^{15}+215a^{13}+85a^{11}-369a^{9}+182a^{7}+102a^{5}-80a^{3}+8a$, $8a^{20}-109a^{18}+578a^{16}-1436a^{14}+1357a^{12}+832a^{10}-2479a^{8}+935a^{6}+759a^{4}-431a^{2}+7$, $3a^{21}-40a^{19}+206a^{17}-489a^{15}+411a^{13}+356a^{11}-817a^{9}+222a^{7}+278a^{5}-117a^{3}-8a+1$, $a-1$, $5a^{21}-69a^{19}+372a^{17}-946a^{15}+937a^{13}+502a^{11}-1678a^{9}+673a^{7}+525a^{5}-298a^{3}-3a-1$, $a^{21}-14a^{19}+77a^{17}-202a^{15}+215a^{13}+85a^{11}-368a^{9}+176a^{7}+110a^{5}-75a^{3}-a+1$, $7a^{21}-96a^{19}+514a^{17}-1298a^{15}+1281a^{13}+662a^{11}-2254a^{9}+961a^{7}+649a^{5}-426a^{3}+18a-1$, $7a^{21}+a^{20}-95a^{19}-13a^{18}+501a^{17}+65a^{16}-1234a^{15}-149a^{14}+1142a^{13}+119a^{12}+748a^{11}+108a^{10}-2116a^{9}-237a^{8}+761a^{7}+70a^{6}+662a^{5}+74a^{4}-359a^{3}-37a^{2}-a+1$, $7a^{21}+a^{20}-96a^{19}-14a^{18}+514a^{17}+77a^{16}-1298a^{15}-202a^{14}+1281a^{13}+215a^{12}+662a^{11}+85a^{10}-2254a^{9}-368a^{8}+961a^{7}+176a^{6}+649a^{5}+110a^{4}-426a^{3}-75a^{2}+18a-1$, $5a^{21}-9a^{20}-69a^{19}+123a^{18}+373a^{17}-655a^{16}-958a^{15}+1638a^{14}+990a^{13}-1573a^{12}+407a^{11}-910a^{10}-1662a^{9}+2833a^{8}+817a^{7}-1109a^{6}+420a^{5}-853a^{4}-347a^{3}+500a^{2}+38a-11$, $5a^{21}+10a^{20}-69a^{19}-135a^{18}+373a^{17}+708a^{16}-959a^{15}-1734a^{14}+1000a^{13}+1596a^{12}+373a^{11}+1041a^{10}-1625a^{9}-2940a^{8}+840a^{7}+1077a^{6}+367a^{5}+891a^{4}-346a^{3}-505a^{2}+51a+12$ (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: | \( 7576192193.94 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{2}\cdot 7576192193.94 \cdot 1}{2\cdot\sqrt{39034415726392643532837005585022976}}\cr\approx \mathstrut & 0.198425189988 \end{aligned}\] (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
11.11.96470357797337.1 |
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: | 22.18.28248324053221546961515534563512015217902284051423179842565864776423831827504888313734987193448503459155287522171306501220244138633403098320586348761480379236352.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.9.0.1}{9} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | $16{,}\,{\href{/padicField/5.6.0.1}{6} }$ | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.14.0.1}{14} }{,}\,{\href{/padicField/11.8.0.1}{8} }$ | ${\href{/padicField/13.11.0.1}{11} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.8.0.1}{8} }$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.5.0.1}{5} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.14.0.1}{14} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.5.0.1}{5} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.5.0.1}{5} }^{2}{,}\,{\href{/padicField/43.4.0.1}{4} }$ | ${\href{/padicField/47.7.0.1}{7} }^{2}{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.9.0.1}{9} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 $22$ | $2$ | $11$ | $22$ | |||
\(137\) | $\Q_{137}$ | $x + 134$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{137}$ | $x + 134$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
137.2.1.2 | $x^{2} + 411$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
137.2.1.2 | $x^{2} + 411$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
137.4.0.1 | $x^{4} + x^{2} + 95 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
137.12.0.1 | $x^{12} + x^{8} + 61 x^{7} + 40 x^{6} + 40 x^{5} + 12 x^{4} + 36 x^{3} + 135 x^{2} + 61 x + 3$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
\(293\) | 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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
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 $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
\(11093\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
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 $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $4$ | $2$ | $2$ | $2$ | ||||
\(216649\) | $\Q_{216649}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{216649}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |