Normalized defining polynomial
\( x^{20} - 4 x^{19} + 16 x^{18} - 44 x^{17} + 112 x^{16} - 232 x^{15} + 440 x^{14} - 713 x^{13} + 1080 x^{12} - 1435 x^{11} + 1778 x^{10} - 1940 x^{9} + 1977 x^{8} - 1748 x^{7} + 1425 x^{6} - 962 x^{5} + 590 x^{4} - 256 x^{3} + 101 x^{2} + 4 x + 1 \)
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: | \(3703216260043975830078125=5^{15}\cdot 3319^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $16.92$ | 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: | $5, 3319$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{124540101436302361} a^{19} - \frac{23672118852482963}{124540101436302361} a^{18} - \frac{14796637364499683}{124540101436302361} a^{17} - \frac{49816177793014618}{124540101436302361} a^{16} + \frac{47669060727944900}{124540101436302361} a^{15} - \frac{45958478007113634}{124540101436302361} a^{14} - \frac{25127361192844954}{124540101436302361} a^{13} + \frac{12399276642419201}{124540101436302361} a^{12} + \frac{3125573156547308}{124540101436302361} a^{11} - \frac{18793364649458049}{124540101436302361} a^{10} - \frac{6600527927328059}{124540101436302361} a^{9} - \frac{45087546193207194}{124540101436302361} a^{8} - \frac{42021453156256220}{124540101436302361} a^{7} - \frac{15556283072363725}{124540101436302361} a^{6} + \frac{26663186304664640}{124540101436302361} a^{5} + \frac{15935898367569040}{124540101436302361} a^{4} + \frac{28312689424854947}{124540101436302361} a^{3} + \frac{52563871394426385}{124540101436302361} a^{2} + \frac{7387064065853802}{124540101436302361} a - \frac{18953434872746568}{124540101436302361}$
Class group and class number
Trivial group, which has order $1$
Unit group
| Rank: | $9$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{7252788353382808}{124540101436302361} a^{19} - \frac{35681897079537674}{124540101436302361} a^{18} + \frac{146272902337002307}{124540101436302361} a^{17} - \frac{430738473553177072}{124540101436302361} a^{16} + \frac{1120787629465618423}{124540101436302361} a^{15} - \frac{2424993578356098792}{124540101436302361} a^{14} + \frac{4684641065239580337}{124540101436302361} a^{13} - \frac{7812220720591282960}{124540101436302361} a^{12} + \frac{11851117709533419067}{124540101436302361} a^{11} - \frac{15938083016860604467}{124540101436302361} a^{10} + \frac{19699421834446635933}{124540101436302361} a^{9} - \frac{21584580433058014574}{124540101436302361} a^{8} + \frac{21650369131016648764}{124540101436302361} a^{7} - \frac{18984538616406360415}{124540101436302361} a^{6} + \frac{15043809057805870870}{124540101436302361} a^{5} - \frac{9857117385470648593}{124540101436302361} a^{4} + \frac{5674729743465996829}{124540101436302361} a^{3} - \frac{2219285512654178219}{124540101436302361} a^{2} + \frac{724064307121078780}{124540101436302361} a + \frac{102342915375914783}{124540101436302361} \) (order $10$) | 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 | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 34590.572989 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$D_5^2:C_4$ (as 20T93):
| A solvable group of order 400 |
| The 28 conjugacy class representatives for $D_5^2:C_4$ |
| Character table for $D_5^2:C_4$ is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 10.6.34424253125.1 |
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 | $20$ | ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ | $20$ | $20$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ | $20$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | $20$ | ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ | $20$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}$ |
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 | ||||||
| 3319 | Data not computed | ||||||