Normalized defining polynomial
\( x^{22} - 1100000000000000000000 x - 10500000000000000000000 \)
Invariants
| Degree: | $22$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[2, 10]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(1304510117015252535175127842044899150362515515004729600000000000000000000=2^{26}\cdot 3^{22}\cdot 5^{20}\cdot 7^{20}\cdot 11^{22}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $1896.60$ | 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: | $2, 3, 5, 7, 11$ | 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 \(a\))
$1$, $a$, $\frac{1}{10} a^{2}$, $\frac{1}{100} a^{3}$, $\frac{1}{1000} a^{4}$, $\frac{1}{10000} a^{5}$, $\frac{1}{100000} a^{6}$, $\frac{1}{1000000} a^{7}$, $\frac{1}{10000000} a^{8}$, $\frac{1}{100000000} a^{9}$, $\frac{1}{1000000000} a^{10}$, $\frac{1}{10000000000} a^{11}$, $\frac{1}{100000000000} a^{12} - \frac{1}{2} a$, $\frac{1}{1000000000000} a^{13} - \frac{1}{20} a^{2}$, $\frac{1}{10000000000000} a^{14} - \frac{1}{200} a^{3}$, $\frac{1}{100000000000000} a^{15} - \frac{1}{2000} a^{4}$, $\frac{1}{1000000000000000} a^{16} - \frac{1}{20000} a^{5}$, $\frac{1}{10000000000000000} a^{17} - \frac{1}{200000} a^{6} - \frac{1}{2} a$, $\frac{1}{100000000000000000} a^{18} - \frac{1}{2000000} a^{7} - \frac{1}{20} a^{2}$, $\frac{1}{500000000000000000} a^{19}$, $\frac{1}{5000000000000000000} a^{20}$, $\frac{1}{350000000000000000000} a^{21} + \frac{3}{35000000000000000000} a^{20} + \frac{1}{1750000000000000000} a^{19} - \frac{1}{350000000000000000} a^{18} + \frac{1}{70000000000000000} a^{17} + \frac{3}{7000000000000000} a^{16} + \frac{1}{350000000000000} a^{15} - \frac{1}{70000000000000} a^{14} - \frac{3}{7000000000000} a^{13} - \frac{1}{350000000000} a^{12} + \frac{1}{70000000000} a^{11} + \frac{3}{7000000000} a^{10} + \frac{1}{350000000} a^{9} - \frac{1}{70000000} a^{8} - \frac{3}{7000000} a^{7} + \frac{3}{1400000} a^{6} - \frac{1}{28000} a^{5} + \frac{3}{7000} a^{4} - \frac{3}{1400} a^{3} + \frac{1}{28} a^{2} + \frac{1}{14} a$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $11$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | 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: | \( 18667522631200000000000000000 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 562000363888803840000 |
| The 513 conjugacy class representatives for t22n58 are not computed |
| Character table for t22n58 is not computed |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | R | R | R | $21{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ | $20{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ | $18{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | $15{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ | ${\href{/LocalNumberField/29.9.0.1}{9} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ | $21{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ | $21{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ | $18{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | $21{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ | ${\href{/LocalNumberField/47.11.0.1}{11} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ | $16{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/59.11.0.1}{11} }{,}\,{\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$ |
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 | Data not computed | ||||||
| 3 | Data not computed | ||||||
| 5 | Data not computed | ||||||
| $7$ | $\Q_{7}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{7}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 7.6.5.2 | $x^{6} - 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
| 7.7.7.6 | $x^{7} + 28 x + 7$ | $7$ | $1$ | $7$ | $F_7$ | $[7/6]_{6}$ | |
| 7.7.8.4 | $x^{7} + 21 x^{2} + 7$ | $7$ | $1$ | $8$ | $F_7$ | $[4/3]_{3}^{2}$ | |
| 11 | Data not computed | ||||||