Normalized defining polynomial
\( x^{20} + 5 x^{18} - 46 x^{16} - 214 x^{14} + 624 x^{12} + 2642 x^{10} - 2143 x^{8} - 8949 x^{6} + 1584 x^{4} + 6536 x^{2} + 43 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[8, 6]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(5834302965409512291058019417402368=2^{10}\cdot 19^{10}\cdot 43^{11}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $48.79$ | 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: | $2, 19, 43$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{19} a^{12} + \frac{3}{19} a^{10} + \frac{6}{19} a^{8} + \frac{7}{19} a^{6} - \frac{6}{19} a^{4} - \frac{9}{19} a^{2} + \frac{5}{19}$, $\frac{1}{19} a^{13} + \frac{3}{19} a^{11} + \frac{6}{19} a^{9} + \frac{7}{19} a^{7} - \frac{6}{19} a^{5} - \frac{9}{19} a^{3} + \frac{5}{19} a$, $\frac{1}{133} a^{14} - \frac{3}{133} a^{10} - \frac{11}{133} a^{8} + \frac{11}{133} a^{6} + \frac{66}{133} a^{4} + \frac{13}{133} a^{2} - \frac{34}{133}$, $\frac{1}{266} a^{15} - \frac{1}{38} a^{12} - \frac{3}{266} a^{11} + \frac{8}{19} a^{10} + \frac{61}{133} a^{9} - \frac{3}{19} a^{8} - \frac{61}{133} a^{7} - \frac{7}{38} a^{6} + \frac{33}{133} a^{5} - \frac{13}{38} a^{4} - \frac{60}{133} a^{3} - \frac{5}{19} a^{2} + \frac{99}{266} a - \frac{5}{38}$, $\frac{1}{3316222} a^{16} + \frac{2}{1658111} a^{14} - \frac{1}{38} a^{13} - \frac{19}{13426} a^{12} + \frac{8}{19} a^{11} + \frac{822009}{1658111} a^{10} - \frac{3}{19} a^{9} - \frac{442812}{1658111} a^{8} - \frac{7}{38} a^{7} - \frac{44822}{1658111} a^{6} - \frac{13}{38} a^{5} + \frac{704013}{1658111} a^{4} - \frac{5}{19} a^{3} + \frac{621443}{3316222} a^{2} - \frac{5}{38} a + \frac{10817}{1658111}$, $\frac{1}{3316222} a^{17} + \frac{2}{1658111} a^{15} - \frac{1}{266} a^{14} - \frac{19}{13426} a^{13} + \frac{822009}{1658111} a^{11} - \frac{65}{133} a^{10} - \frac{442812}{1658111} a^{9} + \frac{11}{266} a^{8} - \frac{44822}{1658111} a^{7} - \frac{11}{266} a^{6} + \frac{704013}{1658111} a^{5} - \frac{33}{133} a^{4} + \frac{621443}{3316222} a^{3} - \frac{13}{266} a^{2} + \frac{10817}{1658111} a + \frac{17}{133}$, $\frac{1}{23213554} a^{18} + \frac{1}{23213554} a^{16} - \frac{4705}{23213554} a^{14} - \frac{1}{38} a^{13} - \frac{1}{1658111} a^{12} - \frac{3}{38} a^{11} + \frac{2065494}{11606777} a^{10} + \frac{13}{38} a^{9} - \frac{2032608}{11606777} a^{8} + \frac{6}{19} a^{7} - \frac{6613597}{23213554} a^{6} - \frac{13}{38} a^{5} - \frac{2630373}{11606777} a^{4} + \frac{9}{38} a^{3} - \frac{8475139}{23213554} a^{2} + \frac{7}{19} a + \frac{4909431}{23213554}$, $\frac{1}{23213554} a^{19} + \frac{1}{23213554} a^{17} - \frac{4705}{23213554} a^{15} - \frac{1}{266} a^{14} - \frac{1}{1658111} a^{13} - \frac{1}{38} a^{12} + \frac{2065494}{11606777} a^{11} + \frac{115}{266} a^{10} - \frac{2032608}{11606777} a^{9} + \frac{51}{133} a^{8} - \frac{6613597}{23213554} a^{7} + \frac{73}{266} a^{6} - \frac{2630373}{11606777} a^{5} + \frac{109}{266} a^{4} - \frac{8475139}{23213554} a^{3} + \frac{25}{133} a^{2} + \frac{4909431}{23213554} a + \frac{66}{133}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $13$ | 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: | \( 2621944424.55 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 10240 |
| The 160 conjugacy class representatives for t20n423 are not computed |
| Character table for t20n423 is not computed |
Intermediate fields
| \(\Q(\sqrt{817}) \), 5.5.667489.1 x5, 10.10.364007458703857.1 |
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 | ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ | R | ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ | R | ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.10.0.1 | $x^{10} - x^{3} + 1$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |
| 2.10.10.9 | $x^{10} - 15 x^{8} + 38 x^{6} - 18 x^{4} + 25 x^{2} - 63$ | $2$ | $5$ | $10$ | $C_2 \times (C_2^4 : C_5)$ | $[2, 2, 2, 2, 2]^{5}$ | |
| $19$ | 19.2.1.1 | $x^{2} - 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 19.2.1.1 | $x^{2} - 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.1 | $x^{2} - 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.1 | $x^{2} - 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.1 | $x^{2} - 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.1 | $x^{2} - 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $43$ | 43.2.1.2 | $x^{2} + 387$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 43.2.1.2 | $x^{2} + 387$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 43.4.3.1 | $x^{4} + 387$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 43.4.2.1 | $x^{4} + 215 x^{2} + 16641$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 43.4.2.1 | $x^{4} + 215 x^{2} + 16641$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 43.4.2.1 | $x^{4} + 215 x^{2} + 16641$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |