Normalized defining polynomial
\( x^{20} - x^{19} + 253 x^{18} - 244 x^{17} + 22278 x^{16} - 1216 x^{15} + 909430 x^{14} + 946310 x^{13} + 18454456 x^{12} + 46632980 x^{11} + 188056520 x^{10} + 901661781 x^{9} + 1033706923 x^{8} + 7259384884 x^{7} + 5672426873 x^{6} + 10581584071 x^{5} + 54459292816 x^{4} - 137803699260 x^{3} + 307969326850 x^{2} - 492455797555 x + 449673743845 \)
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: | \(6026164104418384613756308291011505401611328125=5^{15}\cdot 7^{10}\cdot 31^{18}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $194.54$ | 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: | $5, 7, 31$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1085=5\cdot 7\cdot 31\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1085}(64,·)$, $\chi_{1085}(1,·)$, $\chi_{1085}(643,·)$, $\chi_{1085}(841,·)$, $\chi_{1085}(587,·)$, $\chi_{1085}(659,·)$, $\chi_{1085}(153,·)$, $\chi_{1085}(729,·)$, $\chi_{1085}(27,·)$, $\chi_{1085}(867,·)$, $\chi_{1085}(869,·)$, $\chi_{1085}(678,·)$, $\chi_{1085}(876,·)$, $\chi_{1085}(1007,·)$, $\chi_{1085}(624,·)$, $\chi_{1085}(433,·)$, $\chi_{1085}(946,·)$, $\chi_{1085}(1077,·)$, $\chi_{1085}(281,·)$, $\chi_{1085}(573,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5} a^{16} + \frac{2}{5} a^{15} + \frac{1}{5} a^{12} + \frac{1}{5} a^{10} + \frac{1}{5} a^{8} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4}$, $\frac{1}{5} a^{17} + \frac{1}{5} a^{15} + \frac{1}{5} a^{13} - \frac{2}{5} a^{12} + \frac{1}{5} a^{11} - \frac{2}{5} a^{10} + \frac{1}{5} a^{9} - \frac{2}{5} a^{8} - \frac{2}{5} a^{6} - \frac{2}{5} a^{4}$, $\frac{1}{305} a^{18} - \frac{18}{305} a^{17} + \frac{19}{305} a^{16} + \frac{3}{305} a^{15} + \frac{11}{305} a^{14} - \frac{19}{61} a^{13} - \frac{27}{61} a^{12} + \frac{23}{61} a^{11} - \frac{11}{61} a^{10} + \frac{29}{61} a^{9} - \frac{46}{305} a^{8} - \frac{7}{305} a^{7} - \frac{104}{305} a^{6} + \frac{32}{305} a^{5} + \frac{74}{305} a^{4} - \frac{7}{61} a^{3} + \frac{9}{61} a^{2} + \frac{3}{61} a + \frac{22}{61}$, $\frac{1}{1521495426288915471854008550853136623009340670671103061618943949243391556425803879639771222534614645295533191025} a^{19} - \frac{17844198899395633068453261567844205690879355308656702827341600002745693443225846348086510974529932101577374}{1521495426288915471854008550853136623009340670671103061618943949243391556425803879639771222534614645295533191025} a^{18} - \frac{3402024217296377326160278604484725762490522561254600712956585072263892948049419747709100398036030353099197902}{60859817051556618874160342034125464920373626826844122464757757969735662257032155185590848901384585811821327641} a^{17} - \frac{95674967950119453997831052643796178532518942591811222548239663912927394795469569992966104943620044933628191449}{1521495426288915471854008550853136623009340670671103061618943949243391556425803879639771222534614645295533191025} a^{16} + \frac{113384549322134712780827280616933898059954756720807450096045061972609881712376127833494169665417705563442015216}{304299085257783094370801710170627324601868134134220612323788789848678311285160775927954244506922929059106638205} a^{15} - \frac{16009808831754627734560167539020554170456635326063194636898506890035851960772050053975851105536861692253856426}{1521495426288915471854008550853136623009340670671103061618943949243391556425803879639771222534614645295533191025} a^{14} + \frac{357798274346049618010128116873136322672549948501658463745309934332772892107177169153493000535000481925272556998}{1521495426288915471854008550853136623009340670671103061618943949243391556425803879639771222534614645295533191025} a^{13} + \frac{323402367315229171383336533861954599361151978029619117658562104720370554828800530664870847388761241008033926661}{1521495426288915471854008550853136623009340670671103061618943949243391556425803879639771222534614645295533191025} a^{12} - \frac{611221341453002139919869495367450282958840466538408387226286019807450586928687712654668168005991226248212064762}{1521495426288915471854008550853136623009340670671103061618943949243391556425803879639771222534614645295533191025} a^{11} + \frac{397869335585032582828675426631485396031803538140296943064225534557826001274085953401165313894489445736480375111}{1521495426288915471854008550853136623009340670671103061618943949243391556425803879639771222534614645295533191025} a^{10} + \frac{341412067448842740958332916991893082997448173420940288143809931593419096309270076300528003970270288513988748127}{1521495426288915471854008550853136623009340670671103061618943949243391556425803879639771222534614645295533191025} a^{9} + \frac{127408214674988764479012373142719520150117398572001231278827322933476726025012367317906226074338063536475397858}{304299085257783094370801710170627324601868134134220612323788789848678311285160775927954244506922929059106638205} a^{8} - \frac{211745580865156929419865393934836953559046020557956754280978451692054265148426398230291667336456981730487302482}{1521495426288915471854008550853136623009340670671103061618943949243391556425803879639771222534614645295533191025} a^{7} - \frac{20460503037154308130014425882898685463271272021155835715589220503669881373876403716410979526094722526812386249}{304299085257783094370801710170627324601868134134220612323788789848678311285160775927954244506922929059106638205} a^{6} + \frac{495428194273787640763184666770101591813520847267721139266689534364480009955632789454127831894342043319587887268}{1521495426288915471854008550853136623009340670671103061618943949243391556425803879639771222534614645295533191025} a^{5} + \frac{255322230644620309294386601541514793111518812940135139478691685099669102534629565845854730413806007633217448382}{1521495426288915471854008550853136623009340670671103061618943949243391556425803879639771222534614645295533191025} a^{4} - \frac{116468276728955031847569714143238507438131608866474883277934983211374848560962855061102716988388276455902156131}{304299085257783094370801710170627324601868134134220612323788789848678311285160775927954244506922929059106638205} a^{3} - \frac{44388615844186063553321075086020745042516549254508735842267255554265554298613190980604021188948163858193060244}{304299085257783094370801710170627324601868134134220612323788789848678311285160775927954244506922929059106638205} a^{2} - \frac{69648206547243110552829727515780966567809466348767391053296817629471533855563000993887119817536594580641284228}{304299085257783094370801710170627324601868134134220612323788789848678311285160775927954244506922929059106638205} a - \frac{225899750618928591594831494530227300640173042970893160319768435262194729572503783932775056111805500226953572}{1593188928051220389375925184139410076449571382901678598553868009678944038142203015329603374381795440100034755}$
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{576050}$, which has order $36867200$ (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: | \( 24173706.832424585 \) (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
| \(\Q(\sqrt{5}) \), 4.0.5886125.2, 5.5.923521.1, 10.10.2665284492003125.1 |
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 | $20$ | $20$ | R | R | ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ | $20$ | $20$ | ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ | $20$ | ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ | R | ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ | $20$ | $20$ | $20$ | ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $5$ | 5.4.3.1 | $x^{4} - 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 5.4.3.1 | $x^{4} - 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.4.3.1 | $x^{4} - 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.4.3.1 | $x^{4} - 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.4.3.1 | $x^{4} - 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 7 | Data not computed | ||||||
| $31$ | 31.10.9.1 | $x^{10} - 31$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ |
| 31.10.9.1 | $x^{10} - 31$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ | |