Normalized defining polynomial
\( x^{20} - 10 x^{19} + 17 x^{18} + 122 x^{17} - 413 x^{16} - 412 x^{15} + 2668 x^{14} - 6 x^{13} - 8161 x^{12} + 2452 x^{11} + 13443 x^{10} - 4478 x^{9} - 12172 x^{8} + 2810 x^{7} + 5842 x^{6} - 450 x^{5} - 1317 x^{4} - 84 x^{3} + 104 x^{2} + 20 x + 1 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[20, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(24371997355510472326748292775936=2^{30}\cdot 7^{8}\cdot 13^{14}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $37.10$ | 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: | $2, 7, 13$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{23} a^{17} - \frac{6}{23} a^{16} - \frac{6}{23} a^{15} + \frac{1}{23} a^{14} - \frac{7}{23} a^{13} - \frac{8}{23} a^{12} + \frac{8}{23} a^{11} + \frac{11}{23} a^{10} + \frac{2}{23} a^{9} - \frac{5}{23} a^{8} + \frac{4}{23} a^{7} - \frac{3}{23} a^{6} + \frac{5}{23} a^{5} + \frac{2}{23} a^{4} + \frac{10}{23} a^{3} + \frac{11}{23} a^{2} + \frac{4}{23} a - \frac{1}{23}$, $\frac{1}{23} a^{18} + \frac{4}{23} a^{16} + \frac{11}{23} a^{15} - \frac{1}{23} a^{14} - \frac{4}{23} a^{13} + \frac{6}{23} a^{12} - \frac{10}{23} a^{11} - \frac{1}{23} a^{10} + \frac{7}{23} a^{9} - \frac{3}{23} a^{8} - \frac{2}{23} a^{7} + \frac{10}{23} a^{6} + \frac{9}{23} a^{5} - \frac{1}{23} a^{4} + \frac{2}{23} a^{3} + \frac{1}{23} a^{2} - \frac{6}{23}$, $\frac{1}{1068843606114361} a^{19} + \frac{933655610676}{46471461135407} a^{18} - \frac{2125406832207}{152691943730623} a^{17} + \frac{223396158700886}{1068843606114361} a^{16} + \frac{478301404565779}{1068843606114361} a^{15} + \frac{442860222315802}{1068843606114361} a^{14} - \frac{213299024905508}{1068843606114361} a^{13} - \frac{316453908854660}{1068843606114361} a^{12} + \frac{182593172401595}{1068843606114361} a^{11} + \frac{482664821087496}{1068843606114361} a^{10} - \frac{2926390718444}{97167600555851} a^{9} - \frac{259902413724383}{1068843606114361} a^{8} + \frac{142852425679247}{1068843606114361} a^{7} - \frac{384245515057545}{1068843606114361} a^{6} - \frac{242908764602786}{1068843606114361} a^{5} + \frac{622785587597}{5265239438987} a^{4} - \frac{227952186250440}{1068843606114361} a^{3} + \frac{338973796726020}{1068843606114361} a^{2} + \frac{447881994047374}{1068843606114361} a - \frac{387426056468107}{1068843606114361}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $19$ | 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: | \( 2218512101.18 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times F_5$ (as 20T13):
| A solvable group of order 40 |
| The 10 conjugacy class representatives for $C_2\times F_5$ |
| Character table for $C_2\times F_5$ |
Intermediate fields
| \(\Q(\sqrt{26}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{2}, \sqrt{13})\), 5.5.6889792.1, 10.10.4936800315539456.1, 10.10.379753870426112.1, 10.10.617100039442432.1 |
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 10 siblings: | 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 | R | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ | R | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ | R | ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.4.6.1 | $x^{4} - 6 x^{2} + 4$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ |
| 2.8.12.1 | $x^{8} + 6 x^{6} + 8 x^{5} + 16$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ | |
| 2.8.12.1 | $x^{8} + 6 x^{6} + 8 x^{5} + 16$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ | |
| $7$ | 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.4.2.2 | $x^{4} - 7 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.2 | $x^{4} - 7 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.2 | $x^{4} - 7 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.2 | $x^{4} - 7 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| $13$ | 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 13.8.6.1 | $x^{8} - 13 x^{4} + 2704$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| 13.8.6.1 | $x^{8} - 13 x^{4} + 2704$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |