Normalized defining polynomial
\( x^{22} - 2 x^{21} + 4 x^{20} - 2 x^{18} + 9 x^{17} - 4 x^{16} + 6 x^{15} + 4 x^{14} - 2 x^{13} + 10 x^{12} + 7 x^{10} + 4 x^{9} + 2 x^{8} + 11 x^{7} + x^{6} + 3 x^{5} + 4 x^{4} + 2 x^{3} + 3 x^{2} - x + 1 \)
Invariants
| Degree: | $22$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 11]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-3101889720909924494411835627=-\,3^{11}\cdot 211441^{2}\cdot 625831^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $17.77$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $3, 211441, 625831$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{226307} a^{21} + \frac{58004}{226307} a^{20} + \frac{73859}{226307} a^{19} + \frac{47337}{226307} a^{18} + \frac{47189}{226307} a^{17} + \frac{61978}{226307} a^{16} - \frac{17138}{226307} a^{15} + \frac{59829}{226307} a^{14} + \frac{23133}{226307} a^{13} + \frac{78593}{226307} a^{12} - \frac{88947}{226307} a^{11} - \frac{112696}{226307} a^{10} + \frac{59833}{226307} a^{9} + \frac{28850}{226307} a^{8} - \frac{67163}{226307} a^{7} + \frac{18038}{226307} a^{6} + \frac{94968}{226307} a^{5} - \frac{51183}{226307} a^{4} + \frac{439}{226307} a^{3} - \frac{108055}{226307} a^{2} - \frac{39655}{226307} a - \frac{43583}{226307}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $10$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{174996}{226307} a^{21} - \frac{306194}{226307} a^{20} + \frac{636794}{226307} a^{19} + \frac{44224}{226307} a^{18} - \frac{56186}{226307} a^{17} + \frac{1044341}{226307} a^{16} - \frac{61084}{226307} a^{15} + \frac{873864}{226307} a^{14} + \frac{455466}{226307} a^{13} + \frac{331624}{226307} a^{12} + \frac{931476}{226307} a^{11} + \frac{626913}{226307} a^{10} + \frac{1121234}{226307} a^{9} + \frac{630658}{226307} a^{8} + \frac{676618}{226307} a^{7} + \frac{1405654}{226307} a^{6} + \frac{844504}{226307} a^{5} + \frac{410792}{226307} a^{4} + \frac{331478}{226307} a^{3} + \frac{567526}{226307} a^{2} + \frac{237775}{226307} a + \frac{147846}{226307} \) (order $6$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 73023.7380593 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 79833600 |
| The 112 conjugacy class representatives for t22n47 are not computed |
| Character table for t22n47 is not computed |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 11.5.132326332471.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $22$ | R | $22$ | ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ | $22$ | ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ | $18{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ | ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 3.14.7.2 | $x^{14} + 243 x^{4} - 729 x^{2} + 2187$ | $2$ | $7$ | $7$ | $C_{14}$ | $[\ ]_{2}^{7}$ | |
| 211441 | Data not computed | ||||||
| 625831 | Data not computed | ||||||