Normalized defining polynomial
\( x^{22} - 2 x^{21} - 7 x^{20} + 14 x^{19} + 18 x^{18} - 67 x^{17} - 101 x^{16} + 234 x^{15} + 497 x^{14} - 273 x^{13} - 893 x^{12} + 23 x^{11} + 753 x^{10} + 147 x^{9} - 418 x^{8} - 88 x^{7} + 226 x^{6} + 6 x^{5} - 90 x^{4} + 7 x^{3} + 16 x^{2} - x - 1 \)
Invariants
| Degree: | $22$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[14, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(31919563459480622441144580078125=5^{11}\cdot 43^{2}\cdot 547^{2}\cdot 34374601^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $27.04$ | 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: | $5, 43, 547, 34374601$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{1325914530737524621} a^{21} + \frac{448523034773011435}{1325914530737524621} a^{20} + \frac{60895253381014387}{1325914530737524621} a^{19} - \frac{465043176648431886}{1325914530737524621} a^{18} + \frac{191619929257122309}{1325914530737524621} a^{17} - \frac{247872319115922686}{1325914530737524621} a^{16} + \frac{373802862007984858}{1325914530737524621} a^{15} - \frac{621711033409974498}{1325914530737524621} a^{14} - \frac{518493388111088972}{1325914530737524621} a^{13} - \frac{353055875683546982}{1325914530737524621} a^{12} - \frac{253013172380413357}{1325914530737524621} a^{11} + \frac{629468894051682428}{1325914530737524621} a^{10} + \frac{532727007524532713}{1325914530737524621} a^{9} - \frac{540402043568653027}{1325914530737524621} a^{8} + \frac{125198104722412285}{1325914530737524621} a^{7} - \frac{19433771485550519}{1325914530737524621} a^{6} + \frac{355403583977920202}{1325914530737524621} a^{5} - \frac{427598569834050640}{1325914530737524621} a^{4} + \frac{135729423205129962}{1325914530737524621} a^{3} - \frac{495226434317360913}{1325914530737524621} a^{2} - \frac{386709058999787293}{1325914530737524621} a - \frac{384850908358753902}{1325914530737524621}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $17$ | 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: | \( 73525321.2665 \) (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
| \(\Q(\sqrt{5}) \), 11.7.808524990121.1 |
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 | $22$ | $22$ | R | $22$ | ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/13.14.0.1}{14} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/23.14.0.1}{14} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ | ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | $22$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | R | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 5 | Data not computed | ||||||
| $43$ | 43.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 43.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 43.4.2.1 | $x^{4} + 215 x^{2} + 16641$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 43.14.0.1 | $x^{14} - x + 5$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | |
| 547 | Data not computed | ||||||
| 34374601 | Data not computed | ||||||