Normalized defining polynomial
\( x^{20} - 2 x^{18} - 66 x^{16} + 174 x^{14} + 1245 x^{12} - 3969 x^{10} - 6195 x^{8} + 26542 x^{6} - 10254 x^{4} - 12770 x^{2} + 6385 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[12, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(93878033821762240604784858628096000=2^{20}\cdot 5^{3}\cdot 61^{4}\cdot 397^{4}\cdot 1277^{3}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $56.06$ | 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: | $2, 5, 61, 397, 1277$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{7341947028628841} a^{18} + \frac{3377950413457539}{7341947028628841} a^{16} + \frac{538769107466883}{7341947028628841} a^{14} + \frac{320387970861921}{7341947028628841} a^{12} + \frac{1410521211797855}{7341947028628841} a^{10} + \frac{3633377094197584}{7341947028628841} a^{8} - \frac{3504432381774996}{7341947028628841} a^{6} + \frac{1953389937393555}{7341947028628841} a^{4} + \frac{1433997189563710}{7341947028628841} a^{2} - \frac{642174041445037}{7341947028628841}$, $\frac{1}{7341947028628841} a^{19} + \frac{3377950413457539}{7341947028628841} a^{17} + \frac{538769107466883}{7341947028628841} a^{15} + \frac{320387970861921}{7341947028628841} a^{13} + \frac{1410521211797855}{7341947028628841} a^{11} + \frac{3633377094197584}{7341947028628841} a^{9} - \frac{3504432381774996}{7341947028628841} a^{7} + \frac{1953389937393555}{7341947028628841} a^{5} + \frac{1433997189563710}{7341947028628841} a^{3} - \frac{642174041445037}{7341947028628841} a$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $15$ | 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: | \( 21011820841.9 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 122880 |
| The 252 conjugacy class representatives for t20n791 are not computed |
| Character table for t20n791 is not computed |
Intermediate fields
| 5.5.24217.1, 10.10.3744566823265.1 |
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 | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.8.0.1}{8} }$ | ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ | $16{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ | ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ | $16{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ | $16{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ | ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.10.10.1 | $x^{10} - 9 x^{8} + 54 x^{6} - 38 x^{4} + 41 x^{2} - 17$ | $2$ | $5$ | $10$ | $C_2^4 : C_5$ | $[2, 2, 2, 2]^{5}$ |
| 2.10.10.1 | $x^{10} - 9 x^{8} + 54 x^{6} - 38 x^{4} + 41 x^{2} - 17$ | $2$ | $5$ | $10$ | $C_2^4 : C_5$ | $[2, 2, 2, 2]^{5}$ | |
| $5$ | 5.4.3.3 | $x^{4} + 10$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 5.8.0.1 | $x^{8} + x^{2} - 2 x + 3$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 5.8.0.1 | $x^{8} + x^{2} - 2 x + 3$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| $61$ | 61.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 61.8.0.1 | $x^{8} - x + 17$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 61.8.4.1 | $x^{8} + 14884 x^{4} - 226981 x^{2} + 55383364$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 397 | Data not computed | ||||||
| 1277 | Data not computed | ||||||