Normalized defining polynomial
\( x^{20} - 6 x^{19} + 13 x^{18} - 6 x^{17} - 8 x^{16} - 78 x^{15} + 186 x^{14} + 366 x^{13} - 1637 x^{12} + 1206 x^{11} + 3082 x^{10} - 9678 x^{9} + 39901 x^{8} - 165879 x^{7} + 443766 x^{6} - 798225 x^{5} + 1041604 x^{4} - 1035804 x^{3} + 776719 x^{2} - 392763 x + 100489 \)
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: | \(6348282919278872504865709055049=3^{10}\cdot 401^{10}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $34.68$ | 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: | $3, 401$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois over $\Q$. | |||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{8} + \frac{1}{3}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{9} + \frac{1}{3} a$, $\frac{1}{624816603900789} a^{18} - \frac{28118631694093}{624816603900789} a^{17} + \frac{81341967756829}{624816603900789} a^{16} - \frac{28537681391760}{208272201300263} a^{15} + \frac{4391936362954}{12251305958839} a^{14} + \frac{3848425155664}{208272201300263} a^{13} + \frac{26940755150822}{208272201300263} a^{12} - \frac{99722291590811}{208272201300263} a^{11} + \frac{3785418505054}{32885084415831} a^{10} - \frac{133824098867173}{624816603900789} a^{9} - \frac{182640972146003}{624816603900789} a^{8} - \frac{1914421898656}{12251305958839} a^{7} - \frac{502403087947}{12251305958839} a^{6} + \frac{9142715161527}{208272201300263} a^{5} - \frac{50391781638786}{208272201300263} a^{4} + \frac{45081162629742}{208272201300263} a^{3} - \frac{292065829000472}{624816603900789} a^{2} - \frac{66114450957745}{624816603900789} a - \frac{66461990890}{1971030296217}$, $\frac{1}{22896106605094701942422274150554841} a^{19} + \frac{14824379997539468854}{22896106605094701942422274150554841} a^{18} - \frac{1128804020349701267232469781702815}{22896106605094701942422274150554841} a^{17} + \frac{832227398710024110036044743397726}{22896106605094701942422274150554841} a^{16} + \frac{2426128138652759159608043087820263}{7632035535031567314140758050184947} a^{15} - \frac{179485037221303242086407484294517}{2544011845010522438046919350061649} a^{14} - \frac{3399847110966500812423601594465341}{7632035535031567314140758050184947} a^{13} - \frac{1451729211979203078657940622669297}{7632035535031567314140758050184947} a^{12} - \frac{3352027897966567580098165417406846}{22896106605094701942422274150554841} a^{11} - \frac{7439103578234483071056218727733286}{22896106605094701942422274150554841} a^{10} + \frac{6122970703840033544176054917007331}{22896106605094701942422274150554841} a^{9} - \frac{4606769909813691459815510114420035}{22896106605094701942422274150554841} a^{8} + \frac{214837894196234504140928469883076}{448943266766562783184750473540291} a^{7} + \frac{308835578678127211238465992159090}{7632035535031567314140758050184947} a^{6} - \frac{819636961492315492032868469349892}{7632035535031567314140758050184947} a^{5} - \frac{945674572455922651370870589653244}{2544011845010522438046919350061649} a^{4} - \frac{655281315146606029342289968092658}{1346829800299688349554251420620873} a^{3} + \frac{8143533752311507093734397574720311}{22896106605094701942422274150554841} a^{2} + \frac{8406962646398478817193931950716751}{22896106605094701942422274150554841} a - \frac{7290584257267407954365432038589}{72227465631213570796284776500173}$
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{6}$, which has order $48$ (assuming GRH)
Unit group
| Rank: | $9$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{8753039755363541638}{375594149069618409926027} a^{19} + \frac{3479117607366253925}{1126782447208855229778081} a^{18} + \frac{287980895225363263658}{1126782447208855229778081} a^{17} - \frac{158398212968080431499}{375594149069618409926027} a^{16} - \frac{247307151257197896453}{375594149069618409926027} a^{15} + \frac{681803800536358060026}{375594149069618409926027} a^{14} + \frac{2763855164033023439418}{375594149069618409926027} a^{13} - \frac{4276931886424530272195}{375594149069618409926027} a^{12} - \frac{11319900658976809579678}{375594149069618409926027} a^{11} + \frac{74673925353700443622076}{1126782447208855229778081} a^{10} + \frac{22579863330867452457869}{1126782447208855229778081} a^{9} - \frac{41487399749828130110400}{375594149069618409926027} a^{8} - \frac{138313389415135598057291}{375594149069618409926027} a^{7} - \frac{66317599806381130242614}{375594149069618409926027} a^{6} + \frac{1519418291675262112376655}{375594149069618409926027} a^{5} - \frac{3843969503064654421435900}{375594149069618409926027} a^{4} + \frac{5507691294503434789409557}{375594149069618409926027} a^{3} - \frac{16692038784805555989297265}{1126782447208855229778081} a^{2} + \frac{12529606638683435784622562}{1126782447208855229778081} a - \frac{4682148325374900337470}{1184839586970405078631} \) (order $6$) | 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: | \( 795087.603907 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 20 |
| The 8 conjugacy class representatives for $D_{10}$ |
| Character table for $D_{10}$ |
Intermediate fields
| \(\Q(\sqrt{401}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1203}) \), \(\Q(\sqrt{-3}, \sqrt{401})\), 5.5.160801.1 x5, 10.10.10368641602001.1, 10.0.6283241669043.1 x5, 10.0.2519579909286243.1 x5 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 10 siblings: | 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 | ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 401 | Data not computed | ||||||