Normalized defining polynomial
\( x^{20} + 35 x^{18} + 509 x^{16} + 4020 x^{14} + 18824 x^{12} + 53353 x^{10} + 89335 x^{8} + 82266 x^{6} + 36535 x^{4} + 6770 x^{2} + 401 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 10]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(2745360053836502853259274240000000000=2^{20}\cdot 5^{10}\cdot 401^{9}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $66.36$ | 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: | $2, 5, 401$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3}$, $\frac{1}{3} a^{9} + \frac{1}{3} a$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{6}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{7}$, $\frac{1}{9} a^{16} - \frac{1}{9} a^{8} - \frac{2}{9}$, $\frac{1}{9} a^{17} - \frac{1}{9} a^{9} - \frac{2}{9} a$, $\frac{1}{32339691} a^{18} + \frac{1043516}{32339691} a^{16} + \frac{1512379}{10779897} a^{14} + \frac{1358128}{10779897} a^{12} + \frac{254327}{1702089} a^{10} + \frac{4125262}{32339691} a^{8} + \frac{181006}{10779897} a^{6} - \frac{1819328}{10779897} a^{4} + \frac{8076550}{32339691} a^{2} - \frac{7777177}{32339691}$, $\frac{1}{32339691} a^{19} + \frac{1043516}{32339691} a^{17} + \frac{1512379}{10779897} a^{15} + \frac{1358128}{10779897} a^{13} + \frac{254327}{1702089} a^{11} + \frac{4125262}{32339691} a^{9} + \frac{181006}{10779897} a^{7} - \frac{1819328}{10779897} a^{5} + \frac{8076550}{32339691} a^{3} - \frac{7777177}{32339691} a$
Class group and class number
$C_{2172}$, which has order $2172$ (assuming GRH)
Unit group
| Rank: | $9$ | 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: | \( 2526424.45141 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 40 |
| The 13 conjugacy class representatives for $C_5:D_4$ |
| Character table for $C_5:D_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.0.160400.3, 5.5.160801.1, 10.10.80803005003125.1 |
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 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 | ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ | R | ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
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 | Data not computed | ||||||
| $5$ | 5.10.5.1 | $x^{10} - 50 x^{6} + 625 x^{2} - 12500$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
| 5.10.5.1 | $x^{10} - 50 x^{6} + 625 x^{2} - 12500$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
| 401 | Data not computed | ||||||