Normalized defining polynomial
\( x^{20} - 2 x^{19} - 11 x^{18} + 23 x^{17} + 31 x^{16} - 111 x^{15} + 48 x^{14} + 275 x^{13} - 467 x^{12} - 341 x^{11} + 1168 x^{10} + 107 x^{9} - 1480 x^{8} + 268 x^{7} + 1065 x^{6} - 330 x^{5} - 456 x^{4} + 70 x^{3} + 45 x^{2} - 13 x + 1 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[10, 5]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-4175899735297664052734375=-\,5^{10}\cdot 3359^{5}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $17.02$ | 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: | $5, 3359$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{29} a^{18} + \frac{6}{29} a^{17} + \frac{7}{29} a^{16} - \frac{14}{29} a^{15} - \frac{1}{29} a^{14} + \frac{11}{29} a^{13} - \frac{8}{29} a^{12} - \frac{3}{29} a^{11} + \frac{10}{29} a^{10} + \frac{3}{29} a^{9} - \frac{7}{29} a^{8} - \frac{10}{29} a^{7} + \frac{13}{29} a^{6} + \frac{5}{29} a^{5} - \frac{10}{29} a^{4} - \frac{9}{29} a^{3} + \frac{4}{29} a^{2} - \frac{5}{29} a + \frac{1}{29}$, $\frac{1}{926394383617074089} a^{19} - \frac{12142091053754957}{926394383617074089} a^{18} + \frac{104635434902363236}{926394383617074089} a^{17} - \frac{36124521905853043}{926394383617074089} a^{16} + \frac{63803816984656142}{926394383617074089} a^{15} - \frac{364372664090881846}{926394383617074089} a^{14} + \frac{158384673036536493}{926394383617074089} a^{13} - \frac{398425914069610452}{926394383617074089} a^{12} + \frac{360624798603777909}{926394383617074089} a^{11} + \frac{245098805790538186}{926394383617074089} a^{10} + \frac{187342217621257651}{926394383617074089} a^{9} + \frac{43249910586974174}{926394383617074089} a^{8} + \frac{211127391364622795}{926394383617074089} a^{7} - \frac{116667251704081285}{926394383617074089} a^{6} + \frac{95747215839314037}{926394383617074089} a^{5} + \frac{330894810716138501}{926394383617074089} a^{4} - \frac{170708032847318698}{926394383617074089} a^{3} + \frac{101597223023567945}{926394383617074089} a^{2} + \frac{105729704821201003}{926394383617074089} a - \frac{404105861906535104}{926394383617074089}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $14$ | 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: | \( 55116.0718093 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$D_5\wr C_2$ (as 20T48):
| A solvable group of order 200 |
| The 14 conjugacy class representatives for $D_5\wr C_2$ |
| Character table for $D_5\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.2.83975.2, 10.6.35259003125.1 x5 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 10 siblings: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 25 sibling: | data not computed |
| Degree 40 siblings: | 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 | ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{5}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}$ | ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$ |
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 |
|---|---|---|---|---|---|---|---|
| $5$ | 5.10.5.1 | $x^{10} - 50 x^{6} + 625 x^{2} - 12500$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
| 5.10.5.1 | $x^{10} - 50 x^{6} + 625 x^{2} - 12500$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
| 3359 | Data not computed | ||||||