Normalized defining polynomial
\( x^{22} - 16 x^{20} + 105 x^{18} - 363 x^{16} + 700 x^{14} - 709 x^{12} + 248 x^{10} + 146 x^{8} + \cdots - 1 \)
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: | \(33833812608216375123231853716176896\) \(\medspace = 2^{22}\cdot 2053^{2}\cdot 43747835269^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(37.11\) | 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: | \(2\), \(2053\), \(43747835269\) | 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)
| |
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: | $a$, $a^{21}-16a^{19}+105a^{17}-363a^{15}+700a^{13}-709a^{11}+248a^{9}+146a^{7}-135a^{5}+20a^{3}+4a$, $3a^{20}-46a^{18}+286a^{16}-921a^{14}+1606a^{12}-1365a^{10}+215a^{8}+430a^{6}-206a^{4}-8a^{2}+9$, $a^{19}-14a^{17}+77a^{15}-209a^{13}+282a^{11}-145a^{9}-42a^{7}+62a^{5}-11a^{3}-2a$, $a^{20}-15a^{18}+91a^{16}-286a^{14}+491a^{12}-427a^{10}+103a^{8}+103a^{6}-67a^{4}+a^{2}+3$, $a^{20}-17a^{18}+119a^{16}-439a^{14}+897a^{12}-938a^{10}+291a^{8}+257a^{6}-177a^{4}+a^{2}+8$, $2a^{21}-32a^{19}+210a^{17}-725a^{15}+1388a^{13}-1365a^{11}+394a^{9}+361a^{7}-250a^{5}+10a^{3}+11a$, $3a^{21}-46a^{19}+286a^{17}-921a^{15}+1606a^{13}-1365a^{11}+215a^{9}+430a^{7}-206a^{5}-8a^{3}+10a$, $2a^{20}-31a^{18}+195a^{16}-635a^{14}+1115a^{12}-938a^{10}+112a^{8}+327a^{6}-138a^{4}-12a^{2}+6$, $a^{21}-16a^{19}+106a^{17}-376a^{15}+764a^{13}-854a^{11}+385a^{9}+137a^{7}-179a^{5}+24a^{3}+4a$, $a+1$, $a^{21}-16a^{19}+105a^{17}-363a^{15}+700a^{13}-709a^{11}+248a^{9}+146a^{7}-135a^{5}+20a^{3}+4a-1$, $a^{20}-a^{19}-13a^{18}+13a^{17}+63a^{16}-65a^{15}-132a^{14}+156a^{13}+74a^{12}-179a^{11}+127a^{10}+69a^{9}-154a^{8}+36a^{7}-16a^{6}-38a^{5}+48a^{4}+5a^{3}+a^{2}+3a-2$, $5a^{21}-3a^{20}-78a^{19}+47a^{18}+494a^{17}-299a^{16}-1620a^{15}+985a^{14}+2865a^{13}-1750a^{12}-2430a^{11}+1492a^{10}+302a^{9}-189a^{8}+842a^{7}-522a^{6}-348a^{5}+224a^{4}-38a^{3}+19a^{2}+14a-8$, $3a^{21}-46a^{19}+286a^{17}-921a^{15}+1606a^{13}-1365a^{11}+215a^{9}+430a^{7}-206a^{5}-8a^{3}+9a-1$, $7a^{21}+3a^{20}-110a^{19}-48a^{18}+704a^{17}+314a^{16}-2346a^{15}-1075a^{14}+4264a^{13}+2023a^{12}-3837a^{11}-1919a^{10}+755a^{9}+471a^{8}+1202a^{7}+556a^{6}-646a^{5}-335a^{4}-18a^{3}-a^{2}+33a+15$, $a^{21}+a^{20}-15a^{19}-17a^{18}+91a^{17}+119a^{16}-286a^{15}-439a^{14}+491a^{13}+897a^{12}-427a^{11}-938a^{10}+103a^{9}+291a^{8}+104a^{7}+257a^{6}-73a^{5}-177a^{4}+9a^{3}+a^{2}+3a+8$, $3a^{21}-a^{20}-46a^{19}+15a^{18}+286a^{17}-90a^{16}-921a^{15}+273a^{14}+1606a^{13}-427a^{12}-1365a^{11}+282a^{10}+215a^{9}+34a^{8}+430a^{7}-111a^{6}-206a^{5}+18a^{4}-8a^{3}+7a^{2}+9a$, $a^{21}-14a^{19}+76a^{17}-196a^{15}+218a^{13}-179a^{9}+70a^{7}+39a^{5}-13a^{3}+a^{2}-a$ (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: | \( 6802460318.29 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{2}\cdot 6802460318.29 \cdot 1}{2\cdot\sqrt{33833812608216375123231853716176896}}\cr\approx \mathstrut & 0.191363986764 \end{aligned}\] (assuming GRH)
Galois group
$C_2^{10}.S_{11}$ (as 22T51):
A non-solvable group of order 40874803200 |
The 376 conjugacy class representatives for $C_2^{10}.S_{11}$ |
Character table for $C_2^{10}.S_{11}$ |
Intermediate fields
11.11.89814305807257.1 |
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: | 22.18.12866505116811550175468957469497112065832557397661480997767049137886020632529048773147969332864310325042424932303910127386225872285993080888283393286249352527872.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.11.0.1}{11} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.5.0.1}{5} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.5.0.1}{5} }^{2}$ | ${\href{/padicField/11.11.0.1}{11} }^{2}$ | ${\href{/padicField/13.10.0.1}{10} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.14.0.1}{14} }{,}\,{\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | $18{,}\,{\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.8.0.1}{8} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{6}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.5.0.1}{5} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.7.0.1}{7} }^{2}{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.5.0.1}{5} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 $22$ | $2$ | $11$ | $22$ | |||
\(2053\) | $\Q_{2053}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{2053}$ | $x$ | $1$ | $1$ | $0$ | 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 $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
\(43747835269\) | $\Q_{43747835269}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{43747835269}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{43747835269}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{43747835269}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
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 $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |