Normalized defining polynomial
\( x^{20} - x^{19} + 3 x^{18} - 4 x^{17} + 5 x^{16} - 7 x^{15} + 8 x^{14} - x^{13} + 8 x^{12} - 3 x^{11} + 3 x^{10} + 3 x^{9} - 13 x^{8} + 17 x^{7} + 8 x^{6} - 8 x^{5} + x^{4} + 4 x^{3} + 1 \)
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: | \(362355421972633207561\) \(\medspace = 47^{10}\cdot 83^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(10.66\) | 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: | $47^{1/2}83^{1/2}\approx 62.457985878508765$ | ||
Ramified primes: | \(47\), \(83\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5}a^{18}-\frac{1}{5}a^{17}-\frac{1}{5}a^{16}-\frac{1}{5}a^{14}-\frac{2}{5}a^{13}+\frac{2}{5}a^{12}+\frac{2}{5}a^{11}-\frac{1}{5}a^{9}-\frac{2}{5}a^{8}+\frac{2}{5}a^{7}-\frac{1}{5}a^{5}-\frac{2}{5}a^{4}+\frac{1}{5}a^{3}-\frac{1}{5}a^{2}-\frac{1}{5}$, $\frac{1}{2194608515}a^{19}+\frac{74558294}{2194608515}a^{18}-\frac{1017294346}{2194608515}a^{17}+\frac{6939284}{39901973}a^{16}+\frac{856912424}{2194608515}a^{15}-\frac{994013257}{2194608515}a^{14}-\frac{586393183}{2194608515}a^{13}-\frac{382352308}{2194608515}a^{12}+\frac{9694604}{438921703}a^{11}-\frac{654463201}{2194608515}a^{10}+\frac{509560233}{2194608515}a^{9}+\frac{127710497}{2194608515}a^{8}-\frac{19417716}{438921703}a^{7}+\frac{941830149}{2194608515}a^{6}-\frac{675824012}{2194608515}a^{5}-\frac{811533444}{2194608515}a^{4}-\frac{1050222631}{2194608515}a^{3}+\frac{195415228}{438921703}a^{2}+\frac{339804044}{2194608515}a-\frac{29307092}{438921703}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
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: | $\frac{17167678}{39901973}a^{19}-\frac{198448696}{199509865}a^{18}+\frac{459542381}{199509865}a^{17}-\frac{840311969}{199509865}a^{16}+\frac{260122109}{39901973}a^{15}-\frac{1849657884}{199509865}a^{14}+\frac{2532135297}{199509865}a^{13}-\frac{2478167597}{199509865}a^{12}+\frac{2760348783}{199509865}a^{11}-\frac{578017673}{39901973}a^{10}+\frac{2728324706}{199509865}a^{9}-\frac{1951319303}{199509865}a^{8}+\frac{521684558}{199509865}a^{7}+\frac{368598185}{39901973}a^{6}-\frac{1381145139}{199509865}a^{5}+\frac{302967502}{199509865}a^{4}+\frac{53721484}{199509865}a^{3}+\frac{280222406}{199509865}a^{2}-\frac{56123045}{39901973}a+\frac{294207701}{199509865}$, $\frac{35747331}{438921703}a^{19}-\frac{145326186}{2194608515}a^{18}+\frac{403294346}{2194608515}a^{17}-\frac{16979709}{199509865}a^{16}+\frac{12943137}{438921703}a^{15}+\frac{338271231}{2194608515}a^{14}-\frac{1035587648}{2194608515}a^{13}+\frac{2580202278}{2194608515}a^{12}-\frac{2523174922}{2194608515}a^{11}+\frac{644125245}{438921703}a^{10}-\frac{1462277224}{2194608515}a^{9}+\frac{4268504467}{2194608515}a^{8}-\frac{4522480687}{2194608515}a^{7}+\frac{416180518}{438921703}a^{6}+\frac{2431179411}{2194608515}a^{5}-\frac{9009900908}{2194608515}a^{4}+\frac{6052739344}{2194608515}a^{3}+\frac{2462644731}{2194608515}a^{2}-\frac{267045956}{438921703}a-\frac{225933434}{2194608515}$, $\frac{140164190}{438921703}a^{19}-\frac{3029118987}{2194608515}a^{18}+\frac{5495443702}{2194608515}a^{17}-\frac{1056094673}{199509865}a^{16}+\frac{3543454491}{438921703}a^{15}-\frac{24383649848}{2194608515}a^{14}+\frac{34129248594}{2194608515}a^{13}-\frac{36148677594}{2194608515}a^{12}+\frac{30310058261}{2194608515}a^{11}-\frac{7978649180}{438921703}a^{10}+\frac{33598390482}{2194608515}a^{9}-\frac{25547226321}{2194608515}a^{8}+\frac{7023841916}{2194608515}a^{7}+\frac{6150886199}{438921703}a^{6}-\frac{34791286373}{2194608515}a^{5}-\frac{2801703951}{2194608515}a^{4}+\frac{9638984958}{2194608515}a^{3}+\frac{2035043012}{2194608515}a^{2}-\frac{595370537}{438921703}a+\frac{2421876982}{2194608515}$, $\frac{599914055}{438921703}a^{19}-\frac{6328764117}{2194608515}a^{18}+\frac{14781313397}{2194608515}a^{17}-\frac{2359285278}{199509865}a^{16}+\frac{7763200751}{438921703}a^{15}-\frac{54448405703}{2194608515}a^{14}+\frac{72218992069}{2194608515}a^{13}-\frac{65019050674}{2194608515}a^{12}+\frac{72818839881}{2194608515}a^{11}-\frac{14485773913}{438921703}a^{10}+\frac{68365953842}{2194608515}a^{9}-\frac{42559432106}{2194608515}a^{8}-\frac{5470702269}{2194608515}a^{7}+\frac{14242725327}{438921703}a^{6}-\frac{43343911258}{2194608515}a^{5}-\frac{4552730366}{2194608515}a^{4}+\frac{15215114478}{2194608515}a^{3}+\frac{4852902172}{2194608515}a^{2}-\frac{1309620302}{438921703}a+\frac{5352449662}{2194608515}$, $\frac{5352449662}{2194608515}a^{19}-\frac{8352019937}{2194608515}a^{18}+\frac{22386113103}{2194608515}a^{17}-\frac{658020219}{39901973}a^{16}+\frac{52714386368}{2194608515}a^{15}-\frac{76283151389}{2194608515}a^{14}+\frac{97268002999}{2194608515}a^{13}-\frac{77571441731}{2194608515}a^{12}+\frac{21567729594}{438921703}a^{11}-\frac{88876188867}{2194608515}a^{10}+\frac{88486218551}{2194608515}a^{9}-\frac{52308604856}{2194608515}a^{8}-\frac{5404482700}{438921703}a^{7}+\frac{96462346523}{2194608515}a^{6}-\frac{28394029339}{2194608515}a^{5}+\frac{524313962}{2194608515}a^{4}+\frac{9905180028}{2194608515}a^{3}+\frac{1238936834}{438921703}a^{2}-\frac{4852902172}{2194608515}a+\frac{1309620302}{438921703}$, $\frac{2707494034}{2194608515}a^{19}-\frac{5555157859}{2194608515}a^{18}+\frac{12429096146}{2194608515}a^{17}-\frac{400515119}{39901973}a^{16}+\frac{31963004506}{2194608515}a^{15}-\frac{44760304288}{2194608515}a^{14}+\frac{59543687813}{2194608515}a^{13}-\frac{50880774287}{2194608515}a^{12}+\frac{11611246756}{438921703}a^{11}-\frac{61335888844}{2194608515}a^{10}+\frac{53948495212}{2194608515}a^{9}-\frac{33498872742}{2194608515}a^{8}-\frac{1695935381}{438921703}a^{7}+\frac{61586534701}{2194608515}a^{6}-\frac{26555629888}{2194608515}a^{5}-\frac{15417470496}{2194608515}a^{4}+\frac{8340201046}{2194608515}a^{3}+\frac{2034866506}{438921703}a^{2}-\frac{6407575579}{2194608515}a+\frac{507156570}{438921703}$, $\frac{2529411972}{2194608515}a^{19}-\frac{5532330713}{2194608515}a^{18}+\frac{12279972399}{2194608515}a^{17}-\frac{1995362534}{199509865}a^{16}+\frac{32003869343}{2194608515}a^{15}-\frac{44568041303}{2194608515}a^{14}+\frac{58830908611}{2194608515}a^{13}-\frac{51404782113}{2194608515}a^{12}+\frac{55323097553}{2194608515}a^{11}-\frac{57669217082}{2194608515}a^{10}+\frac{50985256052}{2194608515}a^{9}-\frac{30691585729}{2194608515}a^{8}-\frac{11694648927}{2194608515}a^{7}+\frac{65504322668}{2194608515}a^{6}-\frac{38687198478}{2194608515}a^{5}-\frac{11043805931}{2194608515}a^{4}+\frac{15212489607}{2194608515}a^{3}+\frac{4352691061}{2194608515}a^{2}-\frac{6269809532}{2194608515}a+\frac{3185400421}{2194608515}$, $\frac{3769389187}{2194608515}a^{19}-\frac{5479438171}{2194608515}a^{18}+\frac{14363617537}{2194608515}a^{17}-\frac{2064386926}{199509865}a^{16}+\frac{31905425633}{2194608515}a^{15}-\frac{9127974501}{438921703}a^{14}+\frac{57768211187}{2194608515}a^{13}-\frac{40322926184}{2194608515}a^{12}+\frac{61538631897}{2194608515}a^{11}-\frac{51031028432}{2194608515}a^{10}+\frac{9741648969}{438921703}a^{9}-\frac{23774998318}{2194608515}a^{8}-\frac{25925501378}{2194608515}a^{7}+\frac{65878753533}{2194608515}a^{6}-\frac{622046301}{438921703}a^{5}-\frac{3489893156}{438921703}a^{4}+\frac{5043566969}{2194608515}a^{3}+\frac{13483188839}{2194608515}a^{2}-\frac{4861363692}{2194608515}a+\frac{2302714179}{2194608515}$, $\frac{904504175}{438921703}a^{19}-\frac{7684902578}{2194608515}a^{18}+\frac{19536109453}{2194608515}a^{17}-\frac{2933794162}{199509865}a^{16}+\frac{9416070745}{438921703}a^{15}-\frac{66918843862}{2194608515}a^{14}+\frac{86368648006}{2194608515}a^{13}-\frac{69506968761}{2194608515}a^{12}+\frac{90524548749}{2194608515}a^{11}-\frac{15815417419}{438921703}a^{10}+\frac{75705925198}{2194608515}a^{9}-\frac{42994422364}{2194608515}a^{8}-\frac{23685749551}{2194608515}a^{7}+\frac{18733010240}{438921703}a^{6}-\frac{32577094552}{2194608515}a^{5}-\frac{4564214254}{2194608515}a^{4}+\frac{12290055387}{2194608515}a^{3}+\frac{3849438258}{2194608515}a^{2}-\frac{756814606}{438921703}a+\frac{5327087823}{2194608515}$ | 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: | \( 59.602445011 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 59.602445011 \cdot 1}{2\cdot\sqrt{362355421972633207561}}\cr\approx \mathstrut & 0.15012927302 \end{aligned}\]
Galois group
$C_2\wr D_5$ (as 20T81):
A solvable group of order 320 |
The 20 conjugacy class representatives for $C_2\wr D_5$ |
Character table for $C_2\wr D_5$ |
Intermediate fields
\(\Q(\sqrt{-47}) \), 5.1.2209.1 x5, 10.0.405013523.1, 10.2.19035635581.1, 10.0.229345007.1 |
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 |
Degree 20 siblings: | data not computed |
Degree 32 siblings: | data not computed |
Degree 40 siblings: | data not computed |
Minimal sibling: | 10.0.405013523.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.10.0.1}{10} }^{2}$ | ${\href{/padicField/3.5.0.1}{5} }^{4}$ | ${\href{/padicField/5.4.0.1}{4} }^{4}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.5.0.1}{5} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{6}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{6}$ | ${\href{/padicField/17.5.0.1}{5} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{6}$ | ${\href{/padicField/37.5.0.1}{5} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{10}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{6}$ | R | ${\href{/padicField/53.10.0.1}{10} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(47\) | 47.4.2.1 | $x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
47.4.2.1 | $x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
47.4.2.1 | $x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
47.4.2.1 | $x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
47.4.2.1 | $x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(83\) | $\Q_{83}$ | $x + 81$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{83}$ | $x + 81$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{83}$ | $x + 81$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{83}$ | $x + 81$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{83}$ | $x + 81$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{83}$ | $x + 81$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{83}$ | $x + 81$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{83}$ | $x + 81$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
83.2.1.2 | $x^{2} + 83$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
83.2.0.1 | $x^{2} + 82 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
83.2.0.1 | $x^{2} + 82 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
83.2.0.1 | $x^{2} + 82 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
83.2.0.1 | $x^{2} + 82 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
83.2.1.2 | $x^{2} + 83$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |