Normalized defining polynomial
\( x^{20} - 12 x^{15} + 1538 x^{10} + 44412 x^{5} + 371293 \)
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: | \(17855749972907377862423866455078125=5^{12}\cdot 53^{15}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $51.59$ | 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: | $5, 53$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{4}$, $\frac{1}{24} a^{10} - \frac{5}{24}$, $\frac{1}{24} a^{11} - \frac{5}{24} a$, $\frac{1}{24} a^{12} - \frac{5}{24} a^{2}$, $\frac{1}{24} a^{13} - \frac{5}{24} a^{3}$, $\frac{1}{24} a^{14} - \frac{5}{24} a^{4}$, $\frac{1}{414384} a^{15} + \frac{4339}{414384} a^{10} - \frac{8477}{414384} a^{5} + \frac{41161}{414384}$, $\frac{1}{26934960} a^{16} - \frac{1}{2071920} a^{15} + \frac{1}{60} a^{14} - \frac{1}{60} a^{13} + \frac{1}{120} a^{12} + \frac{263329}{26934960} a^{11} - \frac{4339}{2071920} a^{10} + \frac{1}{10} a^{9} + \frac{1}{10} a^{8} + \frac{1}{10} a^{7} + \frac{820291}{26934960} a^{6} - \frac{405907}{2071920} a^{5} + \frac{5}{12} a^{4} + \frac{23}{60} a^{3} + \frac{11}{24} a^{2} - \frac{7469549}{26934960} a + \frac{787607}{2071920}$, $\frac{1}{350154480} a^{17} - \frac{1}{1035960} a^{15} - \frac{1}{60} a^{14} + \frac{4303573}{350154480} a^{12} + \frac{539}{25899} a^{10} - \frac{1}{5} a^{9} - \frac{1}{5} a^{8} - \frac{4566701}{350154480} a^{7} - \frac{1}{10} a^{6} + \frac{215669}{1035960} a^{5} - \frac{19}{60} a^{4} - \frac{1}{5} a^{3} - \frac{135410609}{350154480} a^{2} + \frac{3}{10} a + \frac{4567}{129495}$, $\frac{1}{4552008240} a^{18} - \frac{1}{2071920} a^{15} - \frac{1}{120} a^{14} - \frac{6142375}{910401648} a^{13} + \frac{1}{60} a^{12} + \frac{1}{120} a^{11} - \frac{12957}{690640} a^{10} - \frac{1}{5} a^{9} + \frac{380603227}{4552008240} a^{8} - \frac{39743}{414384} a^{5} - \frac{43}{120} a^{4} + \frac{449127971}{910401648} a^{3} - \frac{29}{60} a^{2} + \frac{43}{120} a - \frac{301487}{690640}$, $\frac{1}{59176107120} a^{19} + \frac{1}{1035960} a^{15} - \frac{941113523}{59176107120} a^{14} + \frac{1}{60} a^{13} - \frac{1}{60} a^{12} + \frac{1}{120} a^{11} + \frac{4339}{1035960} a^{10} - \frac{2805802541}{59176107120} a^{9} + \frac{1}{10} a^{8} + \frac{1}{10} a^{7} + \frac{1}{10} a^{6} - \frac{215669}{1035960} a^{5} - \frac{4558081093}{11835221424} a^{4} + \frac{5}{12} a^{3} + \frac{23}{60} a^{2} + \frac{11}{24} a + \frac{41161}{1035960}$
Class group and class number
$C_{5}$, which has order $5$ (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: | \( 149186526.14265528 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_4\times F_5$ (as 20T20):
| A solvable group of order 80 |
| The 20 conjugacy class representatives for $C_4\times F_5$ |
| Character table for $C_4\times F_5$ |
Intermediate fields
| \(\Q(\sqrt{53}) \), 4.0.148877.1, 5.1.351125.1, 10.2.6534304578125.1 |
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 | ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ | $20$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ | R | ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$ |
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 | Data not computed | ||||||
| 53 | Data not computed | ||||||