Normalized defining polynomial
\( x^{20} + 164 x^{18} + 10578 x^{16} + 338824 x^{14} + 5621264 x^{12} + 47086368 x^{10} + 203993368 x^{8} + \cdots + 8608032 \)
Invariants
| Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(158322645890088916737377685620382389712556392448\)
\(\medspace = 2^{55}\cdot 41^{19}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(229.07\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
| Galois root discriminant: | $2^{11/4}41^{19/20}\approx 229.07471344056387$ | ||
| Ramified primes: |
\(2\), \(41\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{82}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $20$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(656=2^{4}\cdot 41\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{656}(1,·)$, $\chi_{656}(131,·)$, $\chi_{656}(537,·)$, $\chi_{656}(385,·)$, $\chi_{656}(651,·)$, $\chi_{656}(529,·)$, $\chi_{656}(595,·)$, $\chi_{656}(409,·)$, $\chi_{656}(25,·)$, $\chi_{656}(155,·)$, $\chi_{656}(419,·)$, $\chi_{656}(625,·)$, $\chi_{656}(105,·)$, $\chi_{656}(43,·)$, $\chi_{656}(305,·)$, $\chi_{656}(579,·)$, $\chi_{656}(531,·)$, $\chi_{656}(441,·)$, $\chi_{656}(635,·)$, $\chi_{656}(443,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{512}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{3}a^{3}-\frac{1}{3}a$, $\frac{1}{6}a^{4}+\frac{1}{3}a^{2}$, $\frac{1}{6}a^{5}+\frac{1}{3}a$, $\frac{1}{18}a^{6}+\frac{1}{18}a^{4}-\frac{1}{9}a^{2}$, $\frac{1}{18}a^{7}+\frac{1}{18}a^{5}-\frac{1}{9}a^{3}$, $\frac{1}{36}a^{8}+\frac{2}{9}a^{2}$, $\frac{1}{324}a^{9}+\frac{1}{54}a^{7}+\frac{1}{27}a^{5}-\frac{7}{81}a^{3}-\frac{2}{9}a$, $\frac{1}{324}a^{10}-\frac{1}{108}a^{8}-\frac{1}{54}a^{6}+\frac{2}{81}a^{4}$, $\frac{1}{324}a^{11}-\frac{1}{54}a^{7}+\frac{13}{162}a^{5}-\frac{4}{27}a^{3}+\frac{1}{3}a$, $\frac{1}{1944}a^{12}+\frac{1}{972}a^{10}-\frac{1}{162}a^{8}+\frac{4}{243}a^{6}+\frac{37}{486}a^{4}+\frac{1}{27}a^{2}$, $\frac{1}{1944}a^{13}+\frac{1}{972}a^{11}-\frac{1}{486}a^{7}-\frac{35}{486}a^{5}-\frac{2}{81}a^{3}+\frac{2}{9}a$, $\frac{1}{5832}a^{14}+\frac{1}{5832}a^{12}+\frac{1}{1458}a^{10}+\frac{11}{1458}a^{8}+\frac{1}{729}a^{6}+\frac{49}{729}a^{4}+\frac{14}{81}a^{2}$, $\frac{1}{17496}a^{15}-\frac{1}{8748}a^{13}-\frac{1}{8748}a^{11}+\frac{13}{8748}a^{9}-\frac{11}{2187}a^{7}-\frac{337}{4374}a^{5}-\frac{4}{243}a^{3}+\frac{2}{9}a$, $\frac{1}{2554416}a^{16}+\frac{29}{1277208}a^{14}+\frac{145}{638604}a^{12}+\frac{38}{159651}a^{10}-\frac{2813}{319302}a^{8}-\frac{1177}{159651}a^{6}+\frac{2746}{53217}a^{4}+\frac{866}{5913}a^{2}-\frac{36}{73}$, $\frac{1}{2554416}a^{17}+\frac{29}{1277208}a^{15}+\frac{145}{638604}a^{13}+\frac{38}{159651}a^{11}+\frac{287}{638604}a^{9}-\frac{1177}{159651}a^{7}-\frac{3167}{53217}a^{5}-\frac{10}{5913}a^{3}-\frac{36}{73}a$, $\frac{1}{15\!\cdots\!92}a^{18}-\frac{4330702697}{51\!\cdots\!64}a^{16}-\frac{277512179849}{25\!\cdots\!32}a^{14}-\frac{3157007985851}{38\!\cdots\!48}a^{12}+\frac{19421259882173}{12\!\cdots\!16}a^{10}-\frac{27583516196039}{12\!\cdots\!16}a^{8}+\frac{411670887353963}{19\!\cdots\!74}a^{6}+\frac{85640012345146}{10\!\cdots\!43}a^{4}+\frac{14634796374019}{119383537284027}a^{2}-\frac{649746577081}{1473870830667}$, $\frac{1}{15\!\cdots\!92}a^{19}-\frac{4330702697}{51\!\cdots\!64}a^{17}-\frac{277512179849}{25\!\cdots\!32}a^{15}-\frac{3157007985851}{38\!\cdots\!48}a^{13}+\frac{19421259882173}{12\!\cdots\!16}a^{11}+\frac{6105498115985}{64\!\cdots\!58}a^{9}-\frac{304630336350199}{19\!\cdots\!74}a^{7}+\frac{131485512262283}{21\!\cdots\!86}a^{5}+\frac{17582538035353}{119383537284027}a^{3}+\frac{1489792206980}{4421612492001}a$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
Class group and class number
$C_{2}\times C_{22}\times C_{14346200}$, which has order $631232800$ (assuming GRH)
Unit group
| Rank: | $9$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
| Fundamental units: |
$\frac{2749118581}{12\!\cdots\!16}a^{18}+\frac{592278049703}{17\!\cdots\!88}a^{16}+\frac{777894164482}{358150611852081}a^{14}+\frac{17\!\cdots\!65}{25\!\cdots\!32}a^{12}+\frac{22\!\cdots\!05}{21\!\cdots\!86}a^{10}+\frac{17\!\cdots\!65}{238767074568054}a^{8}+\frac{79\!\cdots\!03}{32\!\cdots\!29}a^{6}+\frac{74\!\cdots\!05}{238767074568054}a^{4}+\frac{225915428862398}{39794512428009}a^{2}-\frac{697912851204}{491290276889}$, $\frac{1853119015}{38\!\cdots\!48}a^{18}+\frac{414649811203}{51\!\cdots\!64}a^{16}+\frac{1728314484419}{32\!\cdots\!29}a^{14}+\frac{699047945443039}{38\!\cdots\!48}a^{12}+\frac{10\!\cdots\!03}{32\!\cdots\!29}a^{10}+\frac{39\!\cdots\!41}{12\!\cdots\!16}a^{8}+\frac{15\!\cdots\!94}{96\!\cdots\!87}a^{6}+\frac{41\!\cdots\!88}{10\!\cdots\!43}a^{4}+\frac{45\!\cdots\!92}{119383537284027}a^{2}+\frac{8518922093918}{1473870830667}$, $\frac{8545234639}{77\!\cdots\!96}a^{18}+\frac{229530870737}{12\!\cdots\!16}a^{16}+\frac{14413647706321}{12\!\cdots\!16}a^{14}+\frac{13\!\cdots\!07}{38\!\cdots\!48}a^{12}+\frac{33\!\cdots\!71}{64\!\cdots\!58}a^{10}+\frac{47\!\cdots\!01}{12\!\cdots\!16}a^{8}+\frac{11\!\cdots\!74}{96\!\cdots\!87}a^{6}+\frac{30\!\cdots\!77}{21\!\cdots\!86}a^{4}+\frac{289182450827426}{119383537284027}a^{2}-\frac{2015339690660}{1473870830667}$, $\frac{3984165863}{25\!\cdots\!32}a^{18}+\frac{5903202019}{235496292724656}a^{16}+\frac{6835736673521}{42\!\cdots\!72}a^{14}+\frac{639412208930395}{12\!\cdots\!16}a^{12}+\frac{16\!\cdots\!27}{21\!\cdots\!86}a^{10}+\frac{25\!\cdots\!61}{42\!\cdots\!72}a^{8}+\frac{977004237080026}{44155554885873}a^{6}+\frac{27\!\cdots\!57}{716301223704162}a^{4}+\frac{1449201654538}{60570034137}a^{2}+\frac{668203024483}{491290276889}$, $\frac{303137}{307303908048}a^{18}+\frac{4047215}{25608659004}a^{16}+\frac{125966507}{12804329502}a^{14}+\frac{5708935952}{19206494253}a^{12}+\frac{113577711853}{25608659004}a^{10}+\frac{192653960444}{6402164751}a^{8}+\frac{1785147017396}{19206494253}a^{6}+\frac{253340742254}{2134054917}a^{4}+\frac{11028647933}{237117213}a^{2}+\frac{7516117}{2927373}$, $\frac{41529718087}{15\!\cdots\!92}a^{18}+\frac{2205708354247}{51\!\cdots\!64}a^{16}+\frac{17031147080707}{64\!\cdots\!58}a^{14}+\frac{15\!\cdots\!57}{19\!\cdots\!74}a^{12}+\frac{14\!\cdots\!03}{12\!\cdots\!16}a^{10}+\frac{10\!\cdots\!25}{12\!\cdots\!16}a^{8}+\frac{26\!\cdots\!76}{96\!\cdots\!87}a^{6}+\frac{11\!\cdots\!47}{21\!\cdots\!86}a^{4}+\frac{71\!\cdots\!06}{119383537284027}a^{2}+\frac{28977443788127}{1473870830667}$, $\frac{309602689121}{15\!\cdots\!92}a^{18}+\frac{16504460154335}{51\!\cdots\!64}a^{16}+\frac{64038532616227}{32\!\cdots\!29}a^{14}+\frac{23\!\cdots\!41}{38\!\cdots\!48}a^{12}+\frac{11\!\cdots\!09}{12\!\cdots\!16}a^{10}+\frac{76\!\cdots\!07}{12\!\cdots\!16}a^{8}+\frac{35\!\cdots\!77}{19\!\cdots\!74}a^{6}+\frac{25\!\cdots\!88}{10\!\cdots\!43}a^{4}+\frac{10\!\cdots\!06}{119383537284027}a^{2}+\frac{6326735928262}{1473870830667}$, $\frac{2401677581}{25\!\cdots\!32}a^{18}+\frac{270087344981}{17\!\cdots\!88}a^{16}+\frac{3019152069383}{28\!\cdots\!48}a^{14}+\frac{920465665098503}{25\!\cdots\!32}a^{12}+\frac{690192404468968}{10\!\cdots\!43}a^{10}+\frac{28\!\cdots\!61}{477534149136108}a^{8}+\frac{89\!\cdots\!08}{32\!\cdots\!29}a^{6}+\frac{42\!\cdots\!19}{716301223704162}a^{4}+\frac{21\!\cdots\!90}{39794512428009}a^{2}+\frac{7849817024215}{491290276889}$, $\frac{70339067435}{51\!\cdots\!64}a^{18}+\frac{473520229489}{21\!\cdots\!86}a^{16}+\frac{119394450309065}{85\!\cdots\!44}a^{14}+\frac{55\!\cdots\!25}{12\!\cdots\!16}a^{12}+\frac{71\!\cdots\!40}{10\!\cdots\!43}a^{10}+\frac{20\!\cdots\!07}{42\!\cdots\!72}a^{8}+\frac{50\!\cdots\!17}{32\!\cdots\!29}a^{6}+\frac{47\!\cdots\!95}{238767074568054}a^{4}+\frac{12\!\cdots\!38}{39794512428009}a^{2}-\frac{2192382654188}{491290276889}$
| sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
| Regulator: | \( 6411717617.202166 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 6411717617.202166 \cdot 631232800}{2\cdot\sqrt{158322645890088916737377685620382389712556392448}}\cr\approx \mathstrut & 487.709385082941 \end{aligned}\] (assuming GRH)
Galois group
| A cyclic group of order 20 |
| The 20 conjugacy class representatives for $C_{20}$ |
| Character table for $C_{20}$ |
Intermediate fields
| \(\Q(\sqrt{82}) \), 4.0.141150208.4, 5.5.2825761.1, 10.10.10727651226221314048.1 |
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 | ${\href{/padicField/3.1.0.1}{1} }^{20}$ | $20$ | $20$ | ${\href{/padicField/11.10.0.1}{10} }^{2}$ | ${\href{/padicField/13.10.0.1}{10} }^{2}$ | $20$ | ${\href{/padicField/19.10.0.1}{10} }^{2}$ | ${\href{/padicField/23.5.0.1}{5} }^{4}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}$ | $20$ | R | $20$ | $20$ | ${\href{/padicField/53.5.0.1}{5} }^{4}$ | $20$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $20$ | $4$ | $5$ | $55$ | |||
|
\(41\)
| 41.20.19.6 | $x^{20} + 369$ | $20$ | $1$ | $19$ | 20T1 | $[\ ]_{20}$ |