Normalized defining polynomial
\( x^{20} - 98 x^{14} + 392 x^{12} + 686 x^{10} + 3773 x^{8} + 52822 x^{4} + 184877 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 10]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(11738285814841942098955993088=2^{20}\cdot 7^{15}\cdot 11^{9}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $25.32$ | 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: | $2, 7, 11$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{7} a^{4}$, $\frac{1}{7} a^{5}$, $\frac{1}{7} a^{6}$, $\frac{1}{7} a^{7}$, $\frac{1}{49} a^{8}$, $\frac{1}{49} a^{9}$, $\frac{1}{196} a^{10} - \frac{1}{28} a^{4} - \frac{1}{2} a^{2} + \frac{1}{4}$, $\frac{1}{196} a^{11} - \frac{1}{28} a^{5} - \frac{1}{2} a^{3} + \frac{1}{4} a$, $\frac{1}{1372} a^{12} + \frac{1}{28} a^{6} - \frac{1}{14} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{1372} a^{13} + \frac{1}{28} a^{7} - \frac{1}{14} a^{5} - \frac{1}{4} a^{3}$, $\frac{1}{1372} a^{14} - \frac{1}{196} a^{8} - \frac{1}{14} a^{6} + \frac{1}{28} a^{4}$, $\frac{1}{1372} a^{15} - \frac{1}{196} a^{9} - \frac{1}{14} a^{7} + \frac{1}{28} a^{5}$, $\frac{1}{9604} a^{16} - \frac{1}{98} a^{8} - \frac{1}{28} a^{6} + \frac{1}{28} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{9604} a^{17} - \frac{1}{98} a^{9} - \frac{1}{28} a^{7} + \frac{1}{28} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a$, $\frac{1}{27819839572} a^{18} + \frac{643105}{13909919786} a^{16} + \frac{418311}{3974262796} a^{14} + \frac{1136783}{3974262796} a^{12} + \frac{70447}{40553702} a^{10} + \frac{1758431}{283875914} a^{8} + \frac{910536}{20276851} a^{6} - \frac{1748983}{81107404} a^{4} - \frac{386215}{2896693} a^{2} - \frac{2321853}{5793386}$, $\frac{1}{27819839572} a^{19} + \frac{643105}{13909919786} a^{17} + \frac{418311}{3974262796} a^{15} + \frac{1136783}{3974262796} a^{13} + \frac{70447}{40553702} a^{11} + \frac{1758431}{283875914} a^{9} + \frac{910536}{20276851} a^{7} - \frac{1748983}{81107404} a^{5} - \frac{386215}{2896693} a^{3} - \frac{2321853}{5793386} a$
Class group and class number
$C_{2}\times C_{2}$, which has order $4$ (assuming GRH)
Unit group
| Rank: | $9$ | 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: | \( 133966.88636361615 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_5\times C_5:D_4$ (as 20T53):
| A solvable group of order 200 |
| The 65 conjugacy class representatives for $C_5\times C_5:D_4$ are not computed |
| Character table for $C_5\times C_5:D_4$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-7}) \), 4.0.60368.1, 10.0.246071287.1 |
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 | R | $20$ | $20$ | R | R | ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ | $20$ | ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{10}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ | $20$ | ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}$ | $20$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.10.10.7 | $x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$ | $2$ | $5$ | $10$ | $C_{10}$ | $[2]^{5}$ |
| 2.10.10.11 | $x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$ | $2$ | $5$ | $10$ | $C_{10}$ | $[2]^{5}$ | |
| 7 | Data not computed | ||||||
| $11$ | 11.10.9.5 | $x^{10} - 8019$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ |
| 11.10.0.1 | $x^{10} + x^{2} - x + 6$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |