Normalized defining polynomial
\( x^{20} - 8 x^{19} + 34 x^{18} - 104 x^{17} + 257 x^{16} - 536 x^{15} + 965 x^{14} - 1526 x^{13} + 2153 x^{12} - 2768 x^{11} + 3257 x^{10} - 3370 x^{9} + 2900 x^{8} - 1990 x^{7} + 1054 x^{6} - 416 x^{5} + 105 x^{4} + 10 x^{3} - 16 x^{2} - 2 x + 5 \)
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: | \(5108307368122015428902912=2^{20}\cdot 17^{5}\cdot 1361^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $17.20$ | 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: | $2, 17, 1361$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{15537615967248371735} a^{19} + \frac{4519404038939923826}{15537615967248371735} a^{18} + \frac{3352009136257785318}{15537615967248371735} a^{17} + \frac{4562656484529421113}{15537615967248371735} a^{16} - \frac{6628883110322315756}{15537615967248371735} a^{15} + \frac{1272955152855942712}{3107523193449674347} a^{14} - \frac{637199709698347010}{3107523193449674347} a^{13} - \frac{4531752193734641116}{15537615967248371735} a^{12} + \frac{2324324647545721644}{15537615967248371735} a^{11} + \frac{289505813781374883}{15537615967248371735} a^{10} + \frac{7081252394474966589}{15537615967248371735} a^{9} + \frac{7628710552620882941}{15537615967248371735} a^{8} - \frac{3863185755882911236}{15537615967248371735} a^{7} + \frac{756095366280760076}{15537615967248371735} a^{6} + \frac{5798994809382748938}{15537615967248371735} a^{5} - \frac{4640340283696340539}{15537615967248371735} a^{4} + \frac{2584224500595520204}{15537615967248371735} a^{3} - \frac{3932681059721716499}{15537615967248371735} a^{2} + \frac{6048957307559997423}{15537615967248371735} a - \frac{970508380334119306}{3107523193449674347}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $9$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{415103875614}{3628042008443} a^{19} - \frac{3040527862050}{3628042008443} a^{18} + \frac{11871859524739}{3628042008443} a^{17} - \frac{33795103520641}{3628042008443} a^{16} + \frac{78487952153436}{3628042008443} a^{15} - \frac{153899822426610}{3628042008443} a^{14} + \frac{259733262647126}{3628042008443} a^{13} - \frac{384172555967670}{3628042008443} a^{12} + \frac{506445918904026}{3628042008443} a^{11} - \frac{612522124634308}{3628042008443} a^{10} + \frac{673815895365242}{3628042008443} a^{9} - \frac{613595620190629}{3628042008443} a^{8} + \frac{413576477661083}{3628042008443} a^{7} - \frac{179966694096807}{3628042008443} a^{6} + \frac{26889928453673}{3628042008443} a^{5} + \frac{20553219532185}{3628042008443} a^{4} - \frac{24238044255074}{3628042008443} a^{3} + \frac{20490599645080}{3628042008443} a^{2} - \frac{2969358337265}{3628042008443} a - \frac{2664086449026}{3628042008443} \) (order $4$) | 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: | \( 45855.5542226 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 5000 |
| The 230 conjugacy class representatives for t20n299 are not computed |
| Character table for t20n299 is not computed |
Intermediate fields
| \(\Q(\sqrt{-1}) \), 4.0.272.1 |
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 | $20$ | ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{5}$ | $20$ | $20$ | ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ | $20$ | ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ | $20$ | ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{5}$ | ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}$ | ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{5}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| $17$ | 17.5.0.1 | $x^{5} - x + 6$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |
| 17.5.0.1 | $x^{5} - x + 6$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 17.10.5.1 | $x^{10} - 578 x^{6} + 83521 x^{2} - 51114852$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
| 1361 | Data not computed | ||||||