Normalized defining polynomial
\( x^{22} + x^{18} - 2x^{16} - 2x^{14} - x^{12} + 3x^{8} + x^{6} + x^{4} - 1 \)
Invariants
Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(147982356498821950109384704\) \(\medspace = 2^{22}\cdot 12917^{2}\cdot 459847^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(15.47\) | 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: | \(2\), \(12917\), \(459847\) | 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: | $11$ | 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: | $a$, $4a^{21}-4a^{19}+7a^{17}-14a^{15}+5a^{13}-6a^{11}+6a^{9}+6a^{7}-3a^{5}+5a^{3}-5a$, $a^{20}+2a^{16}-3a^{14}-4a^{10}+a^{8}+2a^{6}+a^{4}+2a^{2}-1$, $2a^{20}-2a^{18}+4a^{16}-8a^{14}+3a^{12}-5a^{10}+4a^{8}+4a^{6}-a^{4}+4a^{2}-4$, $6a^{20}-5a^{18}+10a^{16}-21a^{14}+5a^{12}-11a^{10}+10a^{8}+11a^{6}-2a^{4}+8a^{2}-8$, $5a^{21}-4a^{19}+9a^{17}-17a^{15}+4a^{13}-10a^{11}+6a^{9}+9a^{7}-a^{5}+8a^{3}-5a-1$, $3a^{21}-2a^{20}-3a^{19}+2a^{18}+5a^{17}-3a^{16}-11a^{15}+7a^{14}+4a^{13}-2a^{12}-4a^{11}+2a^{10}+6a^{9}-3a^{8}+5a^{7}-3a^{6}-3a^{5}+2a^{4}+3a^{3}-2a^{2}-4a+2$, $2a^{21}+a^{20}-2a^{19}-2a^{18}+3a^{17}+2a^{16}-7a^{15}-5a^{14}+2a^{13}+4a^{12}-2a^{11}-a^{10}+3a^{9}+3a^{8}+3a^{7}+a^{6}-2a^{5}-3a^{4}+2a^{3}+2a^{2}-a-2$, $5a^{21}-2a^{20}-4a^{19}+a^{18}+9a^{17}-4a^{16}-17a^{15}+6a^{14}+5a^{13}-a^{12}-10a^{11}+5a^{10}+7a^{9}-a^{8}+8a^{7}-4a^{6}-2a^{5}+8a^{3}-4a^{2}-6a+2$, $2a^{21}+2a^{20}-a^{19}-2a^{18}+4a^{17}+4a^{16}-6a^{15}-8a^{14}+a^{13}+3a^{12}-6a^{11}-5a^{10}+a^{9}+4a^{8}+3a^{7}+4a^{6}+a^{5}+5a^{3}+3a^{2}-2a-4$, $a^{21}-2a^{20}+a^{18}+a^{17}-3a^{16}-2a^{15}+6a^{14}-a^{13}-a^{11}+4a^{10}+a^{9}-2a^{8}+2a^{7}-3a^{6}+a^{3}-3a^{2}+1$ (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: | \( 10254.8559154 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{10}\cdot 10254.8559154 \cdot 1}{2\cdot\sqrt{147982356498821950109384704}}\cr\approx \mathstrut & 0.161678872760 \end{aligned}\] (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
11.1.5939843699.1 |
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: | 22.0.1362025042910688080257818810096245534886848009382349365076221097812844416934017717662457544046594347553335852138496.2 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.11.0.1}{11} }^{2}$ | ${\href{/padicField/5.11.0.1}{11} }^{2}$ | ${\href{/padicField/7.11.0.1}{11} }^{2}$ | ${\href{/padicField/11.9.0.1}{9} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.11.0.1}{11} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.5.0.1}{5} }^{2}{,}\,{\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.9.0.1}{9} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.7.0.1}{7} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.14.0.1}{14} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | $18{,}\,{\href{/padicField/47.4.0.1}{4} }$ | ${\href{/padicField/53.7.0.1}{7} }^{2}{,}\,{\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.10.10.13 | $x^{10} + 10 x^{9} + 10 x^{8} + 56 x^{7} + 192 x^{6} + 800 x^{5} + 1536 x^{4} + 2208 x^{3} + 2224 x^{2} + 96 x - 1056$ | $2$ | $5$ | $10$ | $C_2 \times (C_2^4 : C_5)$ | $[2, 2, 2, 2, 2]^{5}$ |
2.12.12.1 | $x^{12} + 16 x^{11} + 100 x^{10} + 184 x^{9} + 44 x^{8} - 1472 x^{7} - 2336 x^{6} - 1600 x^{5} + 18032 x^{4} + 37504 x^{3} + 66880 x^{2} + 40064 x + 13120$ | $2$ | $6$ | $12$ | 12T134 | $[2, 2, 2, 2, 2, 2]^{6}$ | |
\(12917\) | $\Q_{12917}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{12917}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $4$ | $2$ | $2$ | $2$ | ||||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
\(459847\) | $\Q_{459847}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{459847}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{459847}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{459847}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |