Normalized defining polynomial
\( x^{20} + 260 x^{18} + 27560 x^{16} + 1550575 x^{14} + 50378900 x^{12} + 959638615 x^{10} + 10297284075 x^{8} + 54931371300 x^{6} + 103958469875 x^{4} + 50486565675 x^{2} + 4457372465 \)
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: | \(156206948895540640258789062500000000000000000000=2^{20}\cdot 5^{35}\cdot 13^{15}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $228.92$ | 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: | $2, 5, 13$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1300=2^{2}\cdot 5^{2}\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1300}(129,·)$, $\chi_{1300}(707,·)$, $\chi_{1300}(261,·)$, $\chi_{1300}(1,·)$, $\chi_{1300}(521,·)$, $\chi_{1300}(1227,·)$, $\chi_{1300}(781,·)$, $\chi_{1300}(463,·)$, $\chi_{1300}(1169,·)$, $\chi_{1300}(723,·)$, $\chi_{1300}(203,·)$, $\chi_{1300}(983,·)$, $\chi_{1300}(1243,·)$, $\chi_{1300}(389,·)$, $\chi_{1300}(1041,·)$, $\chi_{1300}(909,·)$, $\chi_{1300}(967,·)$, $\chi_{1300}(649,·)$, $\chi_{1300}(187,·)$, $\chi_{1300}(447,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{13} a^{4}$, $\frac{1}{13} a^{5}$, $\frac{1}{13} a^{6}$, $\frac{1}{91} a^{7} + \frac{1}{7} a$, $\frac{1}{1183} a^{8} - \frac{1}{7} a^{2}$, $\frac{1}{1183} a^{9} - \frac{1}{7} a^{3}$, $\frac{1}{1183} a^{10} + \frac{1}{91} a^{4}$, $\frac{1}{1183} a^{11} + \frac{1}{91} a^{5}$, $\frac{1}{107653} a^{12} - \frac{1}{8281} a^{10} - \frac{1}{8281} a^{8} + \frac{6}{637} a^{6} + \frac{20}{637} a^{4} - \frac{6}{49} a^{2}$, $\frac{1}{107653} a^{13} - \frac{1}{8281} a^{11} - \frac{1}{8281} a^{9} - \frac{1}{637} a^{7} + \frac{20}{637} a^{5} - \frac{6}{49} a^{3} - \frac{1}{7} a$, $\frac{1}{107653} a^{14} + \frac{2}{8281} a^{8} - \frac{15}{49} a^{2}$, $\frac{1}{107653} a^{15} + \frac{2}{8281} a^{9} - \frac{15}{49} a^{3}$, $\frac{1}{981041789} a^{16} + \frac{335}{75464753} a^{14} + \frac{139}{75464753} a^{12} + \frac{2204}{5804981} a^{10} - \frac{2248}{5804981} a^{8} - \frac{11122}{446537} a^{6} + \frac{139}{63791} a^{4} + \frac{1555}{4907} a^{2} + \frac{94}{701}$, $\frac{1}{981041789} a^{17} + \frac{335}{75464753} a^{15} + \frac{139}{75464753} a^{13} + \frac{2204}{5804981} a^{11} - \frac{2248}{5804981} a^{9} - \frac{1308}{446537} a^{7} + \frac{139}{63791} a^{5} + \frac{1555}{4907} a^{3} + \frac{2060}{4907} a$, $\frac{1}{1375599839178444003761} a^{18} - \frac{25244781573}{1375599839178444003761} a^{16} + \frac{118450123831768}{105815372244495692597} a^{14} + \frac{16531659676618}{8139644018807360969} a^{12} + \frac{3024872324109767}{8139644018807360969} a^{10} - \frac{2243591395547335}{8139644018807360969} a^{8} + \frac{9122972857370746}{626126462985181613} a^{6} - \frac{19734509188688251}{626126462985181613} a^{4} - \frac{10146748686794921}{48163574075783201} a^{2} + \frac{146406425655869}{982930083179249}$, $\frac{1}{9629198874249108026327} a^{19} - \frac{1427427496522}{9629198874249108026327} a^{17} - \frac{2317141252034645}{740707605711469848179} a^{15} + \frac{1002938261597372}{740707605711469848179} a^{13} + \frac{5832042119437665}{56977508131651526783} a^{11} + \frac{16635396678526001}{56977508131651526783} a^{9} - \frac{17547944563674183}{4382885240896271291} a^{7} + \frac{101767427427072497}{4382885240896271291} a^{5} - \frac{70624291365260240}{337145018530482407} a^{3} + \frac{997531333629912}{6880510582254743} a$
Class group and class number
$C_{2}\times C_{4}\times C_{19671604}$, which has order $157372832$ (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: | \( 208779686.22504243 \) (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
| \(\Q(\sqrt{65}) \), 4.0.4394000.2, 5.5.390625.1, 10.10.283274078369140625.1 |
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 | $20$ | R | ${\href{/LocalNumberField/7.1.0.1}{1} }^{20}$ | $20$ | R | $20$ | $20$ | $20$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | $20$ | ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ | $20$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ | $20$ | $20$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.10.10.7 | $x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$ | $2$ | $5$ | $10$ | $C_{10}$ | $[2]^{5}$ |
| 2.10.10.7 | $x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$ | $2$ | $5$ | $10$ | $C_{10}$ | $[2]^{5}$ | |
| 5 | Data not computed | ||||||
| 13 | Data not computed | ||||||