Normalized defining polynomial
\( x^{22} - 10 x^{20} + 44 x^{18} - 112 x^{16} + 182 x^{14} - 199 x^{12} + 151 x^{10} - 67 x^{8} - 13 x^{6} + \cdots + 1 \)
Invariants
Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[12, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-10994531599582075769676214829056\) \(\medspace = -\,2^{22}\cdot 1619043113033^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(25.76\) | 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: | \(2\), \(1619043113033\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{13}a^{20}-\frac{2}{13}a^{18}+\frac{2}{13}a^{16}-\frac{5}{13}a^{14}-\frac{1}{13}a^{12}+\frac{1}{13}a^{10}+\frac{3}{13}a^{8}-\frac{4}{13}a^{6}-\frac{6}{13}a^{4}+\frac{1}{13}a^{2}-\frac{5}{13}$, $\frac{1}{13}a^{21}-\frac{2}{13}a^{19}+\frac{2}{13}a^{17}-\frac{5}{13}a^{15}-\frac{1}{13}a^{13}+\frac{1}{13}a^{11}+\frac{3}{13}a^{9}-\frac{4}{13}a^{7}-\frac{6}{13}a^{5}+\frac{1}{13}a^{3}-\frac{5}{13}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $16$ | 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: | $a^{2}-1$, $\frac{37}{13}a^{21}-\frac{282}{13}a^{19}+\frac{906}{13}a^{17}-\frac{1576}{13}a^{15}+\frac{1549}{13}a^{13}-\frac{860}{13}a^{11}+\frac{137}{13}a^{9}+\frac{580}{13}a^{7}-\frac{560}{13}a^{5}-\frac{93}{13}a^{3}+\frac{75}{13}a$, $a^{21}-10a^{19}+44a^{17}-112a^{15}+182a^{13}-199a^{11}+151a^{9}-67a^{7}-13a^{5}+36a^{3}-13a$, $\frac{53}{13}a^{20}-\frac{444}{13}a^{18}+\frac{1614}{13}a^{16}-\frac{3333}{13}a^{14}+\frac{4276}{13}a^{12}-\frac{3639}{13}a^{10}+\frac{2057}{13}a^{8}-\frac{95}{13}a^{6}-\frac{981}{13}a^{4}+\frac{443}{13}a^{2}-\frac{18}{13}$, $a^{19}-8a^{17}+28a^{15}-56a^{13}+70a^{11}-59a^{9}+33a^{7}-a^{5}-15a^{3}+6a$, $\frac{37}{13}a^{21}-\frac{282}{13}a^{19}+\frac{906}{13}a^{17}-\frac{1576}{13}a^{15}+\frac{1549}{13}a^{13}-\frac{860}{13}a^{11}+\frac{137}{13}a^{9}+\frac{580}{13}a^{7}-\frac{560}{13}a^{5}-\frac{80}{13}a^{3}+\frac{62}{13}a$, $\frac{33}{13}a^{21}-\frac{300}{13}a^{19}+\frac{1197}{13}a^{17}-\frac{2752}{13}a^{15}+\frac{4010}{13}a^{13}-\frac{3919}{13}a^{11}+\frac{2621}{13}a^{9}-\frac{769}{13}a^{7}-\frac{666}{13}a^{5}+\frac{683}{13}a^{3}-\frac{152}{13}a$, $\frac{2}{13}a^{20}-\frac{4}{13}a^{18}-\frac{35}{13}a^{16}+\frac{185}{13}a^{14}-\frac{405}{13}a^{12}+\frac{496}{13}a^{10}-\frac{410}{13}a^{8}+\frac{252}{13}a^{6}+\frac{1}{13}a^{4}-\frac{102}{13}a^{2}+\frac{3}{13}$, $a-1$, $\frac{18}{13}a^{21}-\frac{140}{13}a^{19}+\frac{465}{13}a^{17}-\frac{857}{13}a^{15}+\frac{944}{13}a^{13}-\frac{671}{13}a^{11}+\frac{301}{13}a^{9}+\frac{110}{13}a^{7}-\frac{199}{13}a^{5}-\frac{34}{13}a^{3}+\frac{40}{13}a-1$, $\frac{34}{13}a^{21}-\frac{34}{13}a^{20}-\frac{276}{13}a^{19}+\frac{276}{13}a^{18}+\frac{965}{13}a^{17}-\frac{965}{13}a^{16}-\frac{1899}{13}a^{15}+\frac{1899}{13}a^{14}+\frac{2293}{13}a^{13}-\frac{2293}{13}a^{12}-\frac{1825}{13}a^{11}+\frac{1825}{13}a^{10}+\frac{947}{13}a^{9}-\frac{947}{13}a^{8}+\frac{111}{13}a^{7}-\frac{111}{13}a^{6}-\frac{568}{13}a^{5}+\frac{568}{13}a^{4}+\frac{151}{13}a^{3}-\frac{151}{13}a^{2}-\frac{14}{13}a+\frac{1}{13}$, $\frac{16}{13}a^{21}+\frac{17}{13}a^{20}-\frac{110}{13}a^{19}-\frac{151}{13}a^{18}+\frac{305}{13}a^{17}+\frac{593}{13}a^{16}-\frac{418}{13}a^{15}-\frac{1346}{13}a^{14}+\frac{244}{13}a^{13}+\frac{1933}{13}a^{12}+\frac{3}{13}a^{11}-\frac{1855}{13}a^{10}-\frac{121}{13}a^{9}+\frac{1208}{13}a^{8}+\frac{261}{13}a^{7}-\frac{341}{13}a^{6}-\frac{70}{13}a^{5}-\frac{310}{13}a^{4}-\frac{166}{13}a^{3}+\frac{303}{13}a^{2}-\frac{28}{13}a-\frac{20}{13}$, $\frac{20}{13}a^{21}+a^{20}-\frac{144}{13}a^{19}-9a^{18}+\frac{430}{13}a^{17}+35a^{16}-\frac{672}{13}a^{15}-77a^{14}+\frac{539}{13}a^{13}+105a^{12}-\frac{175}{13}a^{11}-94a^{10}-\frac{109}{13}a^{9}+57a^{8}+\frac{362}{13}a^{7}-10a^{6}-\frac{198}{13}a^{5}-23a^{4}-\frac{136}{13}a^{3}+13a^{2}+\frac{43}{13}a$, $\frac{88}{13}a^{21}-\frac{34}{13}a^{20}-\frac{709}{13}a^{19}+\frac{276}{13}a^{18}+\frac{2464}{13}a^{17}-\frac{965}{13}a^{16}-\frac{4821}{13}a^{15}+\frac{1899}{13}a^{14}+\frac{5775}{13}a^{13}-\frac{2293}{13}a^{12}-\frac{4540}{13}a^{11}+\frac{1825}{13}a^{10}+\frac{2292}{13}a^{9}-\frac{947}{13}a^{8}+\frac{350}{13}a^{7}-\frac{111}{13}a^{6}-\frac{1425}{13}a^{5}+\frac{555}{13}a^{4}+\frac{348}{13}a^{3}-\frac{138}{13}a^{2}+\frac{41}{13}a+\frac{1}{13}$, $\frac{41}{13}a^{21}+\frac{47}{13}a^{20}-\frac{316}{13}a^{19}-\frac{393}{13}a^{18}+\frac{1031}{13}a^{17}+\frac{1420}{13}a^{16}-\frac{1830}{13}a^{15}-\frac{2900}{13}a^{14}+\frac{1844}{13}a^{13}+\frac{3658}{13}a^{12}-\frac{1038}{13}a^{11}-\frac{3047}{13}a^{10}+\frac{149}{13}a^{9}+\frac{1688}{13}a^{8}+\frac{681}{13}a^{7}-\frac{19}{13}a^{6}-\frac{688}{13}a^{5}-\frac{854}{13}a^{4}-\frac{63}{13}a^{3}+\frac{307}{13}a^{2}+\frac{133}{13}a-\frac{1}{13}$, $\frac{37}{13}a^{21}-\frac{68}{13}a^{20}-\frac{282}{13}a^{19}+\frac{578}{13}a^{18}+\frac{906}{13}a^{17}-\frac{2138}{13}a^{16}-\frac{1576}{13}a^{15}+\frac{4513}{13}a^{14}+\frac{1549}{13}a^{13}-\frac{5964}{13}a^{12}-\frac{860}{13}a^{11}+\frac{5275}{13}a^{10}+\frac{137}{13}a^{9}-\frac{3168}{13}a^{8}+\frac{580}{13}a^{7}+\frac{441}{13}a^{6}-\frac{560}{13}a^{5}+\frac{1214}{13}a^{4}-\frac{93}{13}a^{3}-\frac{666}{13}a^{2}+\frac{62}{13}a+\frac{67}{13}$ (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: | \( 32432175.4537 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{5}\cdot 32432175.4537 \cdot 1}{2\cdot\sqrt{10994531599582075769676214829056}}\cr\approx \mathstrut & 0.196162961282 \end{aligned}\] (assuming GRH)
Galois group
$C_2^{10}.(C_2\times S_{11})$ (as 22T53):
A non-solvable group of order 81749606400 |
The 752 conjugacy class representatives for $C_2^{10}.(C_2\times S_{11})$ |
Character table for $C_2^{10}.(C_2\times S_{11})$ |
Intermediate fields
11.7.1619043113033.1 |
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 siblings: | 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 | ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.5.0.1}{5} }^{2}$ | $16{,}\,{\href{/padicField/5.6.0.1}{6} }$ | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.7.0.1}{7} }^{2}$ | ${\href{/padicField/11.14.0.1}{14} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.5.0.1}{5} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | $22$ | ${\href{/padicField/29.11.0.1}{11} }^{2}$ | $22$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.10.0.1}{10} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $22$ | $2$ | $11$ | $22$ | |||
\(1619043113033\) | 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 $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |