Normalized defining polynomial
\( x^{20} + 1450 x^{16} - 54665 x^{12} - 48778 x^{10} + 3658350 x^{8} - 21218430 x^{6} + 88410125 x^{4} - 512778725 x^{2} + 512778725 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[4, 8]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(1380670199615709510551840000000000000000=2^{20}\cdot 5^{16}\cdot 29^{15}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $90.57$ | 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: | $2, 5, 29$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{29} a^{4}$, $\frac{1}{29} a^{5}$, $\frac{1}{29} a^{6}$, $\frac{1}{29} a^{7}$, $\frac{1}{841} a^{8}$, $\frac{1}{841} a^{9}$, $\frac{1}{841} a^{10}$, $\frac{1}{841} a^{11}$, $\frac{1}{24389} a^{12}$, $\frac{1}{121945} a^{13} + \frac{2}{4205} a^{11} + \frac{1}{4205} a^{9} + \frac{2}{145} a^{7} + \frac{1}{145} a^{5}$, $\frac{1}{121945} a^{14} - \frac{2}{121945} a^{12} + \frac{1}{4205} a^{10} - \frac{2}{4205} a^{8} + \frac{1}{145} a^{6}$, $\frac{1}{121945} a^{15} + \frac{2}{145} a^{5}$, $\frac{1}{67191695} a^{16} - \frac{6}{2316955} a^{14} + \frac{2}{2316955} a^{12} + \frac{4}{79895} a^{10} - \frac{2}{4205} a^{8} - \frac{2}{145} a^{6} - \frac{8}{551} a^{4} + \frac{4}{19} a^{2} + \frac{9}{19}$, $\frac{1}{67191695} a^{17} - \frac{6}{2316955} a^{15} + \frac{2}{2316955} a^{13} + \frac{4}{79895} a^{11} - \frac{2}{4205} a^{9} - \frac{2}{145} a^{7} - \frac{8}{551} a^{5} + \frac{4}{19} a^{3} + \frac{9}{19} a$, $\frac{1}{71164287309930783205} a^{18} + \frac{179970998024}{71164287309930783205} a^{16} + \frac{2885350193723}{2453940941721751145} a^{14} - \frac{15878390771761}{2453940941721751145} a^{12} - \frac{46352168555802}{84618653162819005} a^{10} + \frac{183712021719}{4453613324358895} a^{8} + \frac{7844505652233}{583576918364269} a^{6} - \frac{5552359185467}{583576918364269} a^{4} + \frac{9526849369643}{20123342012561} a^{2} + \frac{980489047054}{20123342012561}$, $\frac{1}{71164287309930783205} a^{19} + \frac{179970998024}{71164287309930783205} a^{17} + \frac{2885350193723}{2453940941721751145} a^{15} + \frac{848990248160}{490788188344350229} a^{13} - \frac{1221096906136}{16923730632563801} a^{11} + \frac{1242835285538}{4453613324358895} a^{9} - \frac{21147497776518}{2917884591821345} a^{7} - \frac{7638453914774}{2917884591821345} a^{5} + \frac{9526849369643}{20123342012561} a^{3} + \frac{980489047054}{20123342012561} a$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $11$ | 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: | \( 331243139693 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_5:D_5.Q_8$ (as 20T105):
| A solvable group of order 400 |
| The 28 conjugacy class representatives for $C_5:D_5.Q_8$ |
| Character table for $C_5:D_5.Q_8$ is not computed |
Intermediate fields
| \(\Q(\sqrt{29}) \), 4.4.390224.1, 10.2.8012167578125.1 |
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 | $20$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ | $20$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ | $20$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ | R | $20$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ | $20$ | $20$ | $20$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}$ |
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 | ||||||
| $5$ | 5.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| 5.10.10.10 | $x^{10} + 10 x^{8} + 5 x^{6} + 10 x^{5} - 20 x^{4} - 20 x^{2} + 2$ | $5$ | $2$ | $10$ | $F_{5}\times C_2$ | $[5/4]_{4}^{2}$ | |
| 29 | Data not computed | ||||||