Normalized defining polynomial
\( x^{20} - 2 x^{19} + 6 x^{18} - 23 x^{16} + 81 x^{15} - 131 x^{14} + 183 x^{13} - 179 x^{12} + 176 x^{11} - 153 x^{10} + 176 x^{9} - 179 x^{8} + 183 x^{7} - 131 x^{6} + 81 x^{5} - 23 x^{4} + 6 x^{2} - 2 x + 1 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[2, 9]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-84297516676069491318359375=-\,5^{10}\cdot 71^{5}\cdot 263^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $19.78$ | 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: | $5, 71, 263$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{6495341} a^{18} + \frac{1378928}{6495341} a^{17} - \frac{937295}{6495341} a^{16} - \frac{3135075}{6495341} a^{15} - \frac{2381177}{6495341} a^{14} - \frac{445862}{6495341} a^{13} - \frac{99600}{6495341} a^{12} + \frac{3003490}{6495341} a^{11} - \frac{2230686}{6495341} a^{10} - \frac{3183970}{6495341} a^{9} - \frac{2230686}{6495341} a^{8} + \frac{3003490}{6495341} a^{7} - \frac{99600}{6495341} a^{6} - \frac{445862}{6495341} a^{5} - \frac{2381177}{6495341} a^{4} - \frac{3135075}{6495341} a^{3} - \frac{937295}{6495341} a^{2} + \frac{1378928}{6495341} a + \frac{1}{6495341}$, $\frac{1}{266308981} a^{19} - \frac{3}{266308981} a^{18} - \frac{66332333}{266308981} a^{17} + \frac{90493141}{266308981} a^{16} + \frac{120988485}{266308981} a^{15} + \frac{57479061}{266308981} a^{14} + \frac{67780318}{266308981} a^{13} + \frac{967274}{4365721} a^{12} + \frac{49058659}{266308981} a^{11} + \frac{442716}{1142957} a^{10} - \frac{72529589}{266308981} a^{9} - \frac{72529260}{266308981} a^{8} + \frac{103152473}{266308981} a^{7} + \frac{49059021}{266308981} a^{6} + \frac{59003400}{266308981} a^{5} + \frac{67780530}{266308981} a^{4} + \frac{57478957}{266308981} a^{3} + \frac{120988508}{266308981} a^{2} + \frac{90493147}{266308981} a - \frac{66332341}{266308981}$
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: | \( -1 \) (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: | \( 69159.4981123 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 14745600 |
| The 396 conjugacy class representatives for t20n1036 are not computed |
| Character table for t20n1036 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 10.6.1089627903125.1 |
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 | ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ | R | $20$ | ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ | $20$ | ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | $20$ | ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ | $20$ | ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $5$ | 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.12.6.1 | $x^{12} + 500 x^{6} - 3125 x^{2} + 62500$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
| 71 | Data not computed | ||||||
| 263 | Data not computed | ||||||