Normalized defining polynomial
\( x^{22} + 13 x^{20} + 75 x^{18} + 255 x^{16} + 572 x^{14} + 895 x^{12} + 999 x^{10} + 790 x^{8} + 425 x^{6} + 140 x^{4} + 21 x^{2} + 1 \)
Invariants
Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 11]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-73443346282868916473942769664\) \(\medspace = -\,2^{22}\cdot 211441^{2}\cdot 625831^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(20.52\) | 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: | \(2\), \(211441\), \(625831\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $10$ | 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: | $a^{20}+12a^{18}+63a^{16}+192a^{14}+380a^{12}+515a^{10}+484a^{8}+306a^{6}+119a^{4}+21a^{2}$, $a^{18}+11a^{16}+53a^{14}+149a^{12}+274a^{10}+346a^{8}+300a^{6}+170a^{4}+54a^{2}+4$, $a^{16}+10a^{14}+43a^{12}+105a^{10}+161a^{8}+160a^{6}+100a^{4}+34a^{2}+3$, $a^{3}+2a$, $a^{21}+12a^{19}+63a^{17}+192a^{15}+380a^{13}+515a^{11}+484a^{9}+306a^{7}+119a^{5}+21a^{3}$, $2a^{21}+25a^{19}+138a^{17}+447a^{15}+952a^{13}+1410a^{11}+1483a^{9}+1095a^{7}+539a^{5}+153a^{3}+15a$, $a^{21}+12a^{19}+63a^{17}+192a^{15}+380a^{13}+515a^{11}+484a^{9}+305a^{7}+115a^{5}+16a^{3}-3a$, $a^{4}-a^{3}+3a^{2}-2a+2$, $2a^{21}+a^{20}+26a^{19}+13a^{18}+150a^{17}+74a^{16}+509a^{15}+244a^{14}+1134a^{13}+520a^{12}+1748a^{11}+756a^{10}+1900a^{9}+766a^{8}+1438a^{7}+534a^{6}+716a^{5}+241a^{4}+200a^{3}+60a^{2}+17a+5$, $a^{21}+14a^{19}+a^{18}+85a^{17}+10a^{16}+296a^{15}+43a^{14}+662a^{13}+106a^{12}+1010a^{11}+166a^{10}+1079a^{9}+169a^{8}+797a^{7}+109a^{6}+384a^{5}+38a^{4}+103a^{3}+a^{2}+7a-1$ (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: | \( 131820.824712 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{11}\cdot 131820.824712 \cdot 1}{2\cdot\sqrt{73443346282868916473942769664}}\cr\approx \mathstrut & 0.146540055780 \end{aligned}\] (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 $C_2^{10}.(C_2\times S_{11})$ are not computed |
Character table for $C_2^{10}.(C_2\times S_{11})$ is not computed |
Intermediate fields
11.5.132326332471.1 |
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 | ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.7.0.1}{7} }^{2}$ | ${\href{/padicField/5.11.0.1}{11} }^{2}$ | ${\href{/padicField/7.14.0.1}{14} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.6.0.1}{6} }^{2}$ | $16{,}\,{\href{/padicField/13.6.0.1}{6} }$ | ${\href{/padicField/17.11.0.1}{11} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}{,}\,{\href{/padicField/19.6.0.1}{6} }$ | ${\href{/padicField/23.9.0.1}{9} }^{2}{,}\,{\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.5.0.1}{5} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$ | ${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | $20{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.10.0.1}{10} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.4.0.1}{4} }^{3}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{5}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 $22$ | $2$ | $11$ | $22$ | |||
\(211441\) | $\Q_{211441}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{211441}$ | $x$ | $1$ | $1$ | $0$ | 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 $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
\(625831\) | $\Q_{625831}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{625831}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
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 $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |