Normalized defining polynomial
\( x^{22} + 44093388187521 x^{20} + \cdots - 51\!\cdots\!00 \)
Invariants
Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[6, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(465\!\cdots\!736\) \(\medspace = 2^{70}\cdot 3^{22}\cdot 67^{10}\cdot 337^{8}\cdot 2851^{10}\cdot 310501^{8}\cdot 818717^{10}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(2.751\times 10^{9}\) | 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: | \(2\), \(3\), \(67\), \(337\), \(2851\), \(310501\), \(818717\) | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{30}a^{3}-\frac{3}{10}a$, $\frac{1}{46916659556700}a^{4}-\frac{941090456393}{15638886518900}a^{2}$, $\frac{1}{14\!\cdots\!00}a^{5}-\frac{941090456393}{469166595567000}a^{3}+\frac{1}{10}a$, $\frac{1}{22\!\cdots\!00}a^{6}-\frac{941090456393}{73\!\cdots\!00}a^{4}+\frac{2430572857301}{15638886518900}a^{2}$, $\frac{1}{66\!\cdots\!00}a^{7}-\frac{941090456393}{22\!\cdots\!00}a^{5}+\frac{2430572857301}{469166595567000}a^{3}-\frac{2}{5}a$, $\frac{1}{10\!\cdots\!00}a^{8}-\frac{941090456393}{34\!\cdots\!00}a^{6}+\frac{2430572857301}{73\!\cdots\!00}a^{4}-\frac{680268388971}{3909721629725}a^{2}$, $\frac{1}{30\!\cdots\!00}a^{9}-\frac{941090456393}{10\!\cdots\!00}a^{7}+\frac{2430572857301}{22\!\cdots\!00}a^{5}-\frac{226756129657}{39097216297250}a^{3}-\frac{2}{5}a$, $\frac{1}{48\!\cdots\!00}a^{10}-\frac{941090456393}{16\!\cdots\!00}a^{8}+\frac{2430572857301}{34\!\cdots\!00}a^{6}-\frac{226756129657}{61\!\cdots\!00}a^{4}+\frac{668944192819}{3909721629725}a^{2}$, $\frac{1}{48\!\cdots\!00}a^{11}-\frac{941090456393}{16\!\cdots\!00}a^{9}+\frac{2430572857301}{34\!\cdots\!00}a^{7}+\frac{467229115703}{13\!\cdots\!50}a^{5}+\frac{1417247971487}{93833319113400}a^{3}+\frac{1}{10}a$, $\frac{1}{22\!\cdots\!00}a^{12}-\frac{941090456393}{75\!\cdots\!00}a^{10}+\frac{2430572857301}{16\!\cdots\!00}a^{8}-\frac{226756129657}{28\!\cdots\!00}a^{6}+\frac{668944192819}{18\!\cdots\!00}a^{4}-\frac{133838122647}{781944325945}a^{2}$, $\frac{1}{22\!\cdots\!00}a^{13}-\frac{941090456393}{75\!\cdots\!00}a^{11}+\frac{2430572857301}{16\!\cdots\!00}a^{9}+\frac{467229115703}{64\!\cdots\!00}a^{7}+\frac{1417247971487}{44\!\cdots\!00}a^{5}-\frac{5599714501519}{469166595567000}a^{3}-\frac{1}{5}a$, $\frac{1}{10\!\cdots\!00}a^{14}-\frac{941090456393}{35\!\cdots\!00}a^{12}+\frac{2430572857301}{75\!\cdots\!00}a^{10}-\frac{226756129657}{13\!\cdots\!00}a^{8}+\frac{668944192819}{86\!\cdots\!00}a^{6}-\frac{44612707549}{12\!\cdots\!00}a^{4}+\frac{669217431301}{3909721629725}a^{2}$, $\frac{1}{10\!\cdots\!00}a^{15}-\frac{941090456393}{35\!\cdots\!00}a^{13}+\frac{2430572857301}{75\!\cdots\!00}a^{11}+\frac{467229115703}{30\!\cdots\!00}a^{9}+\frac{1417247971487}{20\!\cdots\!00}a^{7}-\frac{5599714501519}{22\!\cdots\!00}a^{5}+\frac{221230650822}{19548608148625}a^{3}+\frac{2}{5}a$, $\frac{1}{50\!\cdots\!00}a^{16}-\frac{941090456393}{16\!\cdots\!00}a^{14}+\frac{2430572857301}{35\!\cdots\!00}a^{12}-\frac{226756129657}{63\!\cdots\!00}a^{10}+\frac{668944192819}{40\!\cdots\!00}a^{8}-\frac{44612707549}{57\!\cdots\!00}a^{6}+\frac{669217431301}{18\!\cdots\!00}a^{4}-\frac{669216079831}{3909721629725}a^{2}$, $\frac{1}{50\!\cdots\!00}a^{17}-\frac{941090456393}{16\!\cdots\!00}a^{15}+\frac{2430572857301}{35\!\cdots\!00}a^{13}-\frac{226756129657}{63\!\cdots\!00}a^{11}-\frac{475722262817}{30\!\cdots\!00}a^{9}-\frac{7089196902427}{10\!\cdots\!00}a^{7}+\frac{5600036318311}{22\!\cdots\!00}a^{5}-\frac{221229975087}{19548608148625}a^{3}$, $\frac{1}{46\!\cdots\!00}a^{18}+\frac{4899265354169}{52\!\cdots\!00}a^{16}-\frac{1768675514749}{41\!\cdots\!00}a^{14}-\frac{1498829676109}{71\!\cdots\!00}a^{12}+\frac{587671138891}{63\!\cdots\!00}a^{10}+\frac{7493655539713}{16\!\cdots\!00}a^{8}-\frac{44075000191}{21\!\cdots\!50}a^{6}-\frac{7493652836773}{73\!\cdots\!00}a^{4}+\frac{3526000020137}{7819443259450}a^{2}-\frac{1}{2}$, $\frac{1}{46\!\cdots\!00}a^{19}+\frac{4899265354169}{52\!\cdots\!00}a^{17}-\frac{1768675514749}{41\!\cdots\!00}a^{15}-\frac{1498829676109}{71\!\cdots\!00}a^{13}+\frac{587671138891}{63\!\cdots\!00}a^{11}+\frac{6842080100239}{48\!\cdots\!00}a^{9}-\frac{4576023116387}{10\!\cdots\!00}a^{7}+\frac{1808383737929}{73\!\cdots\!00}a^{5}-\frac{2284102472193}{156388865189000}a^{3}-\frac{1}{4}a$, $\frac{1}{63\!\cdots\!00}a^{20}+\frac{44\!\cdots\!41}{78\!\cdots\!00}a^{18}-\frac{93\!\cdots\!13}{16\!\cdots\!00}a^{16}-\frac{14\!\cdots\!63}{43\!\cdots\!00}a^{14}-\frac{84\!\cdots\!03}{64\!\cdots\!00}a^{12}-\frac{53\!\cdots\!21}{54\!\cdots\!00}a^{10}-\frac{84\!\cdots\!33}{23\!\cdots\!00}a^{8}-\frac{27\!\cdots\!47}{16\!\cdots\!00}a^{6}-\frac{35\!\cdots\!61}{44\!\cdots\!25}a^{4}-\frac{14\!\cdots\!07}{27\!\cdots\!00}a^{2}-\frac{25\!\cdots\!93}{57\!\cdots\!16}$, $\frac{1}{63\!\cdots\!00}a^{21}+\frac{44\!\cdots\!41}{78\!\cdots\!00}a^{19}-\frac{93\!\cdots\!13}{16\!\cdots\!00}a^{17}-\frac{14\!\cdots\!63}{43\!\cdots\!00}a^{15}-\frac{84\!\cdots\!03}{64\!\cdots\!00}a^{13}-\frac{53\!\cdots\!21}{54\!\cdots\!00}a^{11}-\frac{27\!\cdots\!99}{70\!\cdots\!00}a^{9}-\frac{62\!\cdots\!97}{24\!\cdots\!00}a^{7}-\frac{29\!\cdots\!93}{19\!\cdots\!50}a^{5}-\frac{21\!\cdots\!23}{27\!\cdots\!00}a^{3}-\frac{28\!\cdots\!05}{11\!\cdots\!32}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
not computed
Unit group
Rank: | $13$ | 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: | not computed | 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: | not computed | 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 $
Galois group
$C_2^{10}.M_{11}$ (as 22T43):
A non-solvable group of order 8110080 |
The 52 conjugacy class representatives for $C_2^{10}.M_{11}$ |
Character table for $C_2^{10}.M_{11}$ |
Intermediate fields
11.11.118769262421915560193703211428553337469927424.1 |
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: | 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 | ${\href{/padicField/5.8.0.1}{8} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.11.0.1}{11} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}$ | ${\href{/padicField/13.11.0.1}{11} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.8.0.1}{8} }^{2}{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.8.0.1}{8} }^{2}{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | $16{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.5.0.1}{5} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.6.0.1}{6} }^{3}{,}\,{\href{/padicField/41.4.0.1}{4} }$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | $16{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.11.0.1}{11} }^{2}$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
2.4.4.4 | $x^{4} - 2 x^{3} + 4 x^{2} + 12 x + 12$ | $2$ | $2$ | $4$ | $D_{4}$ | $[2, 2]^{2}$ | |
Deg $16$ | $16$ | $1$ | $64$ | ||||
\(3\) | 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $18$ | $9$ | $2$ | $20$ | ||||
\(67\) | 67.2.1.2 | $x^{2} + 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
67.2.1.2 | $x^{2} + 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
67.2.0.1 | $x^{2} + 63 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
67.8.4.2 | $x^{8} + 35912 x^{4} - 16241202 x^{2} + 40302242$ | $2$ | $4$ | $4$ | $C_8$ | $[\ ]_{2}^{4}$ | |
67.8.4.1 | $x^{8} + 284 x^{6} + 108 x^{5} + 28074 x^{4} - 13608 x^{3} + 1141144 x^{2} - 1396332 x + 15837397$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(337\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $8$ | $2$ | $4$ | $4$ | ||||
Deg $8$ | $2$ | $4$ | $4$ | ||||
\(2851\) | $\Q_{2851}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{2851}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $8$ | $2$ | $4$ | $4$ | ||||
Deg $8$ | $2$ | $4$ | $4$ | ||||
\(310501\) | Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $4$ | $2$ | $2$ | $2$ | ||||
Deg $6$ | $2$ | $3$ | $3$ | ||||
Deg $6$ | $2$ | $3$ | $3$ | ||||
\(818717\) | 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 $8$ | $2$ | $4$ | $4$ | ||||
Deg $8$ | $2$ | $4$ | $4$ |