Normalized defining polynomial
\( x^{20} + 520 x^{18} + 110240 x^{16} + 12404600 x^{14} + 806062400 x^{12} + 30708435680 x^{10} + 659026180800 x^{8} + 7031215526400 x^{6} + 26613368288000 x^{4} + 25849121625600 x^{2} + 4564349404160 \)
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: | \(159955915669033615625000000000000000000000000000000=2^{30}\cdot 5^{35}\cdot 13^{15}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $323.74$ | 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: | \(2600=2^{3}\cdot 5^{2}\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{2600}(1,·)$, $\chi_{2600}(1227,·)$, $\chi_{2600}(129,·)$, $\chi_{2600}(521,·)$, $\chi_{2600}(203,·)$, $\chi_{2600}(1169,·)$, $\chi_{2600}(707,·)$, $\chi_{2600}(1243,·)$, $\chi_{2600}(1689,·)$, $\chi_{2600}(1561,·)$, $\chi_{2600}(2267,·)$, $\chi_{2600}(2081,·)$, $\chi_{2600}(1763,·)$, $\chi_{2600}(1041,·)$, $\chi_{2600}(2209,·)$, $\chi_{2600}(2283,·)$, $\chi_{2600}(723,·)$, $\chi_{2600}(1747,·)$, $\chi_{2600}(649,·)$, $\chi_{2600}(187,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{52} a^{4}$, $\frac{1}{52} a^{5}$, $\frac{1}{104} a^{6}$, $\frac{1}{728} a^{7} + \frac{1}{7} a$, $\frac{1}{18928} a^{8} - \frac{1}{14} a^{2}$, $\frac{1}{18928} a^{9} - \frac{1}{14} a^{3}$, $\frac{1}{37856} a^{10} + \frac{1}{364} a^{4}$, $\frac{1}{37856} a^{11} + \frac{1}{364} a^{5}$, $\frac{1}{6889792} a^{12} - \frac{1}{264992} a^{10} - \frac{1}{132496} a^{8} + \frac{3}{2548} a^{6} + \frac{5}{637} a^{4} - \frac{3}{49} a^{2}$, $\frac{1}{6889792} a^{13} - \frac{1}{264992} a^{11} - \frac{1}{132496} a^{9} - \frac{1}{5096} a^{7} + \frac{5}{637} a^{5} - \frac{3}{49} a^{3} - \frac{1}{7} a$, $\frac{1}{13779584} a^{14} + \frac{1}{66248} a^{8} - \frac{15}{98} a^{2}$, $\frac{1}{13779584} a^{15} + \frac{1}{66248} a^{9} - \frac{15}{98} a^{3}$, $\frac{1}{251146697984} a^{16} + \frac{335}{9659488384} a^{14} + \frac{139}{4829744192} a^{12} + \frac{551}{46439848} a^{10} - \frac{281}{11609962} a^{8} - \frac{5561}{1786148} a^{6} + \frac{139}{255164} a^{4} + \frac{1555}{9814} a^{2} + \frac{94}{701}$, $\frac{1}{251146697984} a^{17} + \frac{335}{9659488384} a^{15} + \frac{139}{4829744192} a^{13} + \frac{551}{46439848} a^{11} - \frac{281}{11609962} a^{9} - \frac{327}{893074} a^{7} + \frac{139}{255164} a^{5} + \frac{1555}{9814} a^{3} + \frac{2060}{4907} a$, $\frac{1}{704307117659363329925632} a^{18} - \frac{25244781573}{352153558829681664962816} a^{16} + \frac{14806265478971}{1693045955911931081552} a^{14} + \frac{8265829838309}{260468608601835551008} a^{12} + \frac{3024872324109767}{260468608601835551008} a^{10} - \frac{2243591395547335}{130234304300917775504} a^{8} + \frac{4561486428685373}{2504505851940726452} a^{6} - \frac{19734509188688251}{2504505851940726452} a^{4} - \frac{10146748686794921}{96327148151566402} a^{2} + \frac{146406425655869}{982930083179249}$, $\frac{1}{4930149823615543309479424} a^{19} - \frac{713713748261}{1232537455903885827369856} a^{17} - \frac{2317141252034645}{94810573531068140566912} a^{15} + \frac{250734565399343}{11851321691383517570864} a^{13} + \frac{5832042119437665}{1823280260212848857056} a^{11} + \frac{16635396678526001}{911640130106424428528} a^{9} - \frac{17547944563674183}{35063081927170170328} a^{7} + \frac{101767427427072497}{17531540963585085164} a^{5} - \frac{35312145682630120}{337145018530482407} a^{3} + \frac{997531333629912}{6880510582254743} a$
Class group and class number
$C_{2}\times C_{190}\times C_{12845900}$, which has order $4881442000$ (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
| \(\Q(\sqrt{65}) \), 4.0.17576000.2, 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 | $20$ | $20$ | $20$ | ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ | $20$ | ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ | $20$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ | $20$ | $20$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.10.15.9 | $x^{10} - 6 x^{8} - 24 x^{6} + 80 x^{4} + 336 x^{2} + 33056$ | $2$ | $5$ | $15$ | $C_{10}$ | $[3]^{5}$ |
| 2.10.15.9 | $x^{10} - 6 x^{8} - 24 x^{6} + 80 x^{4} + 336 x^{2} + 33056$ | $2$ | $5$ | $15$ | $C_{10}$ | $[3]^{5}$ | |
| 5 | Data not computed | ||||||
| 13 | Data not computed | ||||||