Normalized defining polynomial
\( x^{22} - 12 x^{20} + 59 x^{18} - 151 x^{16} + 204 x^{14} - 106 x^{12} - 64 x^{10} + 101 x^{8} - 19 x^{6} - 16 x^{4} + 3 x^{2} + 1 \)
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: | \(-237871803919090816246272099352576\) \(\medspace = -\,2^{22}\cdot 29^{2}\cdot 131^{2}\cdot 5399^{2}\cdot 367163^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(29.62\) | 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: | \(2\), \(29\), \(131\), \(5399\), \(367163\) | 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: | $16$ | 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: | $a$, $a^{21}-11a^{19}+48a^{17}-103a^{15}+101a^{13}-5a^{11}-69a^{9}+32a^{7}+13a^{5}-3a^{3}$, $a^{3}-2a$, $3a^{21}-38a^{19}+201a^{17}-571a^{15}+915a^{13}-735a^{11}+51a^{9}+381a^{7}-228a^{5}+3a^{3}+18a$, $2a^{21}-25a^{19}+130a^{17}-361a^{15}+559a^{13}-417a^{11}-15a^{9}+249a^{7}-124a^{5}-10a^{3}+11a$, $a^{20}-11a^{18}+49a^{16}-112a^{14}+132a^{12}-55a^{10}-37a^{8}+41a^{6}-6a^{4}$, $2a^{20}-24a^{18}+117a^{16}-293a^{14}+378a^{12}-170a^{10}-137a^{8}+166a^{6}-15a^{4}-18a^{2}-1$, $a^{16}-10a^{14}+39a^{12}-73a^{10}+58a^{8}+9a^{6}-40a^{4}+10a^{2}+5$, $a^{20}-12a^{18}+59a^{16}-151a^{14}+205a^{12}-113a^{10}-47a^{8}+85a^{6}-19a^{4}-6a^{2}+1$, $a+1$, $a^{21}-a^{20}-11a^{19}+10a^{18}+48a^{17}-37a^{16}-103a^{15}+54a^{14}+101a^{13}+11a^{12}-5a^{11}-126a^{10}-69a^{9}+117a^{8}+32a^{7}+21a^{6}+13a^{5}-66a^{4}-3a^{3}+6a^{2}+7$, $4a^{21}+2a^{20}-47a^{19}-22a^{18}+225a^{17}+97a^{16}-557a^{15}-215a^{14}+722a^{13}+232a^{12}-354a^{11}-53a^{10}-211a^{9}-123a^{8}+304a^{7}+89a^{6}-58a^{5}+6a^{4}-25a^{3}-9a^{2}+5a$, $a^{2}-a-1$, $a^{21}-2a^{20}-11a^{19}+23a^{18}+49a^{17}-107a^{16}-112a^{15}+254a^{14}+132a^{13}-305a^{12}-55a^{11}+112a^{10}-37a^{9}+127a^{8}+41a^{7}-122a^{6}-6a^{5}+a^{4}+14a^{2}+a+1$, $a^{20}-11a^{18}+49a^{16}-112a^{14}+131a^{12}-48a^{10}-54a^{8}+57a^{6}-7a^{4}-6a^{2}+a$, $a^{21}-11a^{19}+3a^{18}+49a^{17}-32a^{16}-112a^{15}+137a^{14}+131a^{13}-297a^{12}-48a^{11}+321a^{10}-54a^{9}-93a^{8}+57a^{7}-135a^{6}-6a^{5}+111a^{4}-9a^{3}-6a^{2}+a-9$ (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: | \( 117361710.917 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{5}\cdot 117361710.917 \cdot 1}{2\cdot\sqrt{237871803919090816246272099352576}}\cr\approx \mathstrut & 0.152610439526 \end{aligned}\] (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
11.9.7530807227563.1 |
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 | ${\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.10.0.1}{10} }$ | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}{,}\,{\href{/padicField/11.6.0.1}{6} }$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | $16{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | $18{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.5.0.1}{5} }^{2}$ | ${\href{/padicField/41.11.0.1}{11} }^{2}$ | $16{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | $22$ | $16{,}\,{\href{/padicField/53.6.0.1}{6} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.6.6.1 | $x^{6} + 6 x^{5} + 34 x^{4} + 80 x^{3} + 204 x^{2} + 216 x + 216$ | $2$ | $3$ | $6$ | $A_4$ | $[2, 2]^{3}$ |
Deg $16$ | $2$ | $8$ | $16$ | ||||
\(29\) | 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
29.4.0.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
29.4.0.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
29.5.0.1 | $x^{5} + 3 x + 27$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
29.5.0.1 | $x^{5} + 3 x + 27$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
\(131\) | 131.2.1.2 | $x^{2} + 131$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
131.2.1.2 | $x^{2} + 131$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
131.4.0.1 | $x^{4} + 9 x^{2} + 109 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
131.4.0.1 | $x^{4} + 9 x^{2} + 109 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
131.10.0.1 | $x^{10} + 124 x^{5} + 97 x^{4} + 9 x^{3} + 126 x^{2} + 44 x + 2$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
\(5399\) | 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 $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
\(367163\) | Deg $4$ | $2$ | $2$ | $2$ | |||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |