Normalized defining polynomial
\( x^{20} + 30 x^{18} + 435 x^{16} - 11 x^{15} + 3750 x^{14} - 265 x^{13} + 20825 x^{12} - 2735 x^{11} + 74975 x^{10} - 15000 x^{9} + 173240 x^{8} - 42800 x^{7} + 238320 x^{6} - 63648 x^{5} + 172800 x^{4} - 28800 x^{3} + 32000 x^{2} + 2560 x + 1024 \)
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: | \(164422235102392733097076416015625=5^{34}\cdot 7^{10}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $40.81$ | 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: | $5, 7$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(175=5^{2}\cdot 7\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{175}(64,·)$, $\chi_{175}(1,·)$, $\chi_{175}(69,·)$, $\chi_{175}(6,·)$, $\chi_{175}(71,·)$, $\chi_{175}(139,·)$, $\chi_{175}(76,·)$, $\chi_{175}(141,·)$, $\chi_{175}(146,·)$, $\chi_{175}(29,·)$, $\chi_{175}(34,·)$, $\chi_{175}(99,·)$, $\chi_{175}(36,·)$, $\chi_{175}(134,·)$, $\chi_{175}(104,·)$, $\chi_{175}(41,·)$, $\chi_{175}(106,·)$, $\chi_{175}(174,·)$, $\chi_{175}(111,·)$, $\chi_{175}(169,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{13} - \frac{1}{4} a^{11} + \frac{3}{8} a^{9} - \frac{3}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{8} a^{6} + \frac{1}{8} a^{5} + \frac{1}{8} a^{4} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{14} - \frac{1}{8} a^{12} + \frac{3}{16} a^{10} + \frac{5}{16} a^{9} + \frac{3}{8} a^{8} + \frac{7}{16} a^{7} - \frac{7}{16} a^{6} + \frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{32} a^{15} - \frac{1}{16} a^{13} + \frac{3}{32} a^{11} - \frac{11}{32} a^{10} + \frac{3}{16} a^{9} - \frac{9}{32} a^{8} + \frac{9}{32} a^{7} + \frac{1}{32} a^{6} - \frac{1}{32} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3}$, $\frac{1}{64} a^{16} - \frac{1}{32} a^{14} + \frac{3}{64} a^{12} - \frac{11}{64} a^{11} - \frac{13}{32} a^{10} + \frac{23}{64} a^{9} + \frac{9}{64} a^{8} + \frac{1}{64} a^{7} + \frac{31}{64} a^{6} - \frac{1}{8} a^{5} - \frac{3}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{128} a^{17} - \frac{1}{64} a^{15} + \frac{3}{128} a^{13} - \frac{11}{128} a^{12} - \frac{13}{64} a^{11} - \frac{41}{128} a^{10} - \frac{55}{128} a^{9} - \frac{63}{128} a^{8} - \frac{33}{128} a^{7} - \frac{1}{16} a^{6} - \frac{3}{16} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{38656} a^{18} - \frac{11}{9664} a^{17} + \frac{85}{19328} a^{16} + \frac{17}{2416} a^{15} - \frac{357}{38656} a^{14} - \frac{1695}{38656} a^{13} - \frac{233}{19328} a^{12} + \frac{9427}{38656} a^{11} - \frac{7611}{38656} a^{10} - \frac{823}{38656} a^{9} - \frac{1529}{38656} a^{8} - \frac{2133}{4832} a^{7} + \frac{2777}{9664} a^{6} - \frac{761}{4832} a^{5} + \frac{767}{2416} a^{4} - \frac{157}{604} a^{3} - \frac{2}{151} a^{2} + \frac{27}{151} a - \frac{38}{151}$, $\frac{1}{29317056377047323534965427990016} a^{19} + \frac{116287832883067636192753635}{14658528188523661767482713995008} a^{18} - \frac{47519750953249340144151147983}{14658528188523661767482713995008} a^{17} - \frac{9125333208894981157810533657}{7329264094261830883741356997504} a^{16} + \frac{215644833076811151846496236459}{29317056377047323534965427990016} a^{15} - \frac{687635327532069781048445878153}{29317056377047323534965427990016} a^{14} + \frac{11074425076886542504964237651}{1832316023565457720935339249376} a^{13} - \frac{2801539483297529393290296536089}{29317056377047323534965427990016} a^{12} - \frac{6819036550099762160637523191805}{29317056377047323534965427990016} a^{11} - \frac{6603318871590759902366637395501}{29317056377047323534965427990016} a^{10} - \frac{10377461095959016213927154511263}{29317056377047323534965427990016} a^{9} - \frac{3583111321607642088603252303129}{14658528188523661767482713995008} a^{8} + \frac{3308080390916865441669819604155}{7329264094261830883741356997504} a^{7} - \frac{1221045298687900354990066780361}{3664632047130915441870678498752} a^{6} + \frac{239124181401389628723774818269}{1832316023565457720935339249376} a^{5} - \frac{421778264695122252693282485319}{916158011782728860467669624688} a^{4} + \frac{113534101401828514441400276225}{229039502945682215116917406172} a^{3} - \frac{13729286036256328330940188693}{114519751472841107558458703086} a^{2} - \frac{51414517201724033813935762467}{114519751472841107558458703086} a + \frac{13903602718528357874238958125}{57259875736420553779229351543}$
Class group and class number
$C_{4}\times C_{4}\times C_{4}\times C_{4}$, which has order $256$ (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: | \( 161406.837641 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times C_{10}$ (as 20T3):
| An abelian group of order 20 |
| The 20 conjugacy class representatives for $C_2\times C_{10}$ |
| Character table for $C_2\times C_{10}$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{5}, \sqrt{-7})\), 5.5.390625.1, \(\Q(\zeta_{25})^+\), 10.0.12822723388671875.1, 10.0.2564544677734375.1 |
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 | ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ | R | R | ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 5 | Data not computed | ||||||
| $7$ | 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |