Normalized defining polynomial
\( x^{20} - 4 x^{19} - x^{18} + 28 x^{17} - 60 x^{16} + 45 x^{15} + 65 x^{14} - 215 x^{13} + 233 x^{12} - x^{11} - 318 x^{10} + 407 x^{9} - 158 x^{8} - 175 x^{7} + 270 x^{6} - 75 x^{5} - 110 x^{4} + 46 x^{3} + 24 x^{2} - 2 x - 1 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[10, 5]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-4936006137688415732421875=-\,5^{10}\cdot 11^{17}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $17.17$ | 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: | $5, 11$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{468238596550385959} a^{19} + \frac{31561617846187413}{468238596550385959} a^{18} + \frac{232319717662641602}{468238596550385959} a^{17} - \frac{64048574701025375}{468238596550385959} a^{16} + \frac{75851900668974160}{468238596550385959} a^{15} + \frac{54987615982025592}{468238596550385959} a^{14} + \frac{14361844544542553}{468238596550385959} a^{13} - \frac{105122907236779295}{468238596550385959} a^{12} - \frac{116973137940060106}{468238596550385959} a^{11} - \frac{201538677102207954}{468238596550385959} a^{10} - \frac{70249049007198466}{468238596550385959} a^{9} + \frac{64936547287265046}{468238596550385959} a^{8} - \frac{57757511490261294}{468238596550385959} a^{7} + \frac{221348321038386811}{468238596550385959} a^{6} + \frac{4166937864215002}{468238596550385959} a^{5} - \frac{56068451653526706}{468238596550385959} a^{4} + \frac{173531938015225787}{468238596550385959} a^{3} + \frac{121599202523473067}{468238596550385959} a^{2} + \frac{203102775176949614}{468238596550385959} a + \frac{91388427023054715}{468238596550385959}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $14$ | 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: | \( 63613.3081038 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_5\times D_4$ (as 20T12):
| A solvable group of order 40 |
| The 25 conjugacy class representatives for $C_5\times D_4$ |
| Character table for $C_5\times D_4$ is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.2.275.1, \(\Q(\zeta_{11})^+\), 10.10.669871503125.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | data not computed |
| Degree 20 sibling: | data not computed |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $20$ | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ | R | $20$ | R | $20$ | $20$ | ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$ |
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 | Data not computed | ||||||
| $11$ | 11.10.9.1 | $x^{10} - 11$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ |
| 11.10.8.5 | $x^{10} - 2321 x^{5} + 2033647$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ | |