Normalized defining polynomial
\( x^{20} + 680 x^{18} + 181560 x^{16} + 24969600 x^{14} + 1951674800 x^{12} + 89321484320 x^{10} + 2339342636800 x^{8} + 31990547504000 x^{6} + 183353317216000 x^{4} + 332169297779200 x^{2} + 17454472483840 \)
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: | \(8945072035968174353125000000000000000000000000000000=2^{30}\cdot 5^{35}\cdot 17^{15}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $395.89$ | 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: | $2, 5, 17$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(3400=2^{3}\cdot 5^{2}\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{3400}(1,·)$, $\chi_{3400}(3013,·)$, $\chi_{3400}(2721,·)$, $\chi_{3400}(2889,·)$, $\chi_{3400}(973,·)$, $\chi_{3400}(3277,·)$, $\chi_{3400}(1361,·)$, $\chi_{3400}(1237,·)$, $\chi_{3400}(2041,·)$, $\chi_{3400}(2333,·)$, $\chi_{3400}(293,·)$, $\chi_{3400}(2209,·)$, $\chi_{3400}(2597,·)$, $\chi_{3400}(849,·)$, $\chi_{3400}(681,·)$, $\chi_{3400}(557,·)$, $\chi_{3400}(1653,·)$, $\chi_{3400}(169,·)$, $\chi_{3400}(1529,·)$, $\chi_{3400}(1917,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{68} a^{4}$, $\frac{1}{68} a^{5}$, $\frac{1}{136} a^{6}$, $\frac{1}{952} a^{7} + \frac{2}{7} a$, $\frac{1}{32368} a^{8} + \frac{3}{14} a^{2}$, $\frac{1}{32368} a^{9} + \frac{3}{14} a^{3}$, $\frac{1}{64736} a^{10} + \frac{1}{238} a^{4}$, $\frac{1}{64736} a^{11} + \frac{1}{238} a^{5}$, $\frac{1}{2201024} a^{12} + \frac{3}{952} a^{6}$, $\frac{1}{2201024} a^{13} + \frac{1}{7} a$, $\frac{1}{215700352} a^{14} - \frac{1}{15407168} a^{12} + \frac{1}{226576} a^{10} + \frac{11}{1586032} a^{8} - \frac{5}{3332} a^{6} - \frac{6}{833} a^{4} - \frac{11}{343} a^{2} - \frac{3}{7}$, $\frac{1}{215700352} a^{15} - \frac{1}{15407168} a^{13} + \frac{1}{226576} a^{11} + \frac{11}{1586032} a^{9} - \frac{3}{6664} a^{7} - \frac{6}{833} a^{5} - \frac{11}{343} a^{3} - \frac{1}{7} a$, $\frac{1}{740715008768} a^{16} - \frac{3}{2723216944} a^{14} + \frac{1}{22884176} a^{12} - \frac{2451}{320378464} a^{10} + \frac{345}{160189232} a^{8} - \frac{2071}{673064} a^{6} - \frac{6325}{1177862} a^{4} + \frac{1767}{69286} a^{2} + \frac{233}{707}$, $\frac{1}{740715008768} a^{17} - \frac{3}{2723216944} a^{15} + \frac{1}{22884176} a^{13} - \frac{2451}{320378464} a^{11} + \frac{345}{160189232} a^{9} + \frac{25}{336532} a^{7} - \frac{6325}{1177862} a^{5} + \frac{1767}{69286} a^{3} + \frac{132}{707} a$, $\frac{1}{244942982005569563616334336} a^{18} - \frac{8337329642421}{15308936375348097726020896} a^{16} + \frac{7016538840383035}{3602102676552493582593152} a^{14} - \frac{246162546423674863}{1801051338276246791296576} a^{12} - \frac{129429256266235671}{52972098184595493861664} a^{10} + \frac{100797270551658313}{13243024546148873465416} a^{8} + \frac{25019729895427101}{45823614346535894344} a^{6} + \frac{255314247292880572}{97375180486388775481} a^{4} - \frac{1341950508756782847}{5727951793316986793} a^{2} + \frac{39466456267633324}{116896975373816057}$, $\frac{1}{1714600874038986945314340352} a^{19} - \frac{8337329642421}{107162554627436684082146272} a^{17} - \frac{2420757267540529}{6303679683966863769538016} a^{15} - \frac{2584102053899996003}{12607359367933727539076032} a^{13} + \frac{522903318959067353}{92701171823042114257912} a^{11} + \frac{1236367888468728581}{92701171823042114257912} a^{9} - \frac{42252493273003511}{5453010107237771426936} a^{7} - \frac{7087674904712282561}{1363252526809442856734} a^{5} - \frac{6407904661565134367}{80191325106437815102} a^{3} + \frac{89565159999268777}{818278827616712399} a$
Class group and class number
$C_{2}\times C_{7684068100}$, which has order $15368136200$ (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: | \( 423245425.59528744 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 20 |
| The 20 conjugacy class representatives for $C_{20}$ |
| Character table for $C_{20}$ |
Intermediate fields
| \(\Q(\sqrt{85}) \), 4.0.39304000.1, 5.5.390625.1, 10.10.1083264923095703125.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/7.1.0.1}{1} }^{20}$ | $20$ | $20$ | R | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ | $20$ | $20$ | ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ | $20$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ | $20$ | $20$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$ |
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 | Data not computed | ||||||
| 17 | Data not computed | ||||||