Normalized defining polynomial
\( x^{20} - 294 x^{15} - 64864 x^{10} + 5348406 x^{5} + 28629151 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[4, 8]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(120780545291490852832794189453125=5^{31}\cdot 11^{10}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $40.19$ | 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: | $5, 11$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{4}$, $\frac{1}{300} a^{10} - \frac{6}{25} a^{5} - \frac{149}{300}$, $\frac{1}{300} a^{11} - \frac{6}{25} a^{6} - \frac{149}{300} a$, $\frac{1}{300} a^{12} - \frac{6}{25} a^{7} - \frac{149}{300} a^{2}$, $\frac{1}{300} a^{13} - \frac{6}{25} a^{8} - \frac{149}{300} a^{3}$, $\frac{1}{300} a^{14} - \frac{6}{25} a^{9} - \frac{149}{300} a^{4}$, $\frac{1}{4474601400} a^{15} + \frac{2506507}{4474601400} a^{10} - \frac{666863537}{4474601400} a^{5} + \frac{798408229}{4474601400}$, $\frac{1}{138712643400} a^{16} - \frac{33966953}{46237547800} a^{11} - \frac{3999428317}{27742528680} a^{6} + \frac{3960168121}{46237547800} a$, $\frac{1}{4300091945400} a^{17} - \frac{2345844343}{1433363981800} a^{12} + \frac{151358732291}{860018389080} a^{7} - \frac{437608413369}{1433363981800} a^{2}$, $\frac{1}{666514251537000} a^{18} + \frac{1}{10750229863500} a^{17} - \frac{1}{346781608500} a^{16} - \frac{1}{22373007000} a^{15} - \frac{88347683251}{222171417179000} a^{13} - \frac{7102176469}{5375114931750} a^{12} + \frac{166544299}{173390804250} a^{11} - \frac{2506507}{22373007000} a^{10} - \frac{75268831933217}{666514251537000} a^{8} - \frac{1952264264147}{10750229863500} a^{7} + \frac{3351624377}{346781608500} a^{6} + \frac{5141464937}{22373007000} a^{5} - \frac{92258905087477}{222171417179000} a^{3} + \frac{745024971346}{2687557465875} a^{2} - \frac{23163738737}{173390804250} a + \frac{3676193171}{22373007000}$, $\frac{1}{20661941797647000} a^{19} - \frac{1}{21500459727000} a^{17} - \frac{1}{346781608500} a^{16} + \frac{1}{11186503500} a^{15} - \frac{6753490198621}{6887313932549000} a^{14} - \frac{21629746607}{21500459727000} a^{12} + \frac{166544299}{173390804250} a^{11} + \frac{2506507}{11186503500} a^{10} + \frac{3697201831766203}{20661941797647000} a^{9} - \frac{842795500363}{21500459727000} a^{7} + \frac{3351624377}{346781608500} a^{6} + \frac{451786813}{11186503500} a^{5} - \frac{2987152470929847}{6887313932549000} a^{4} + \frac{7734295878571}{21500459727000} a^{2} - \frac{23163738737}{173390804250} a + \frac{1917058579}{11186503500}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $11$ | 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: | \( 184304774.0008127 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times F_5$ (as 20T9):
| 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
| \(\Q(\sqrt{5}) \), 4.4.15125.1, 5.1.78125.1, 10.2.30517578125.1 |
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 | ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ | R | ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$ |
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 | Data not computed | ||||||
| $11$ | 11.10.5.2 | $x^{10} + 1331 x^{4} - 14641 x^{2} + 805255$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
| 11.10.5.2 | $x^{10} + 1331 x^{4} - 14641 x^{2} + 805255$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |