Invariants
| Base field: | $\F_{193}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 27 x + 193 x^{2} )( 1 - 25 x + 193 x^{2} )$ |
| $1 - 52 x + 1061 x^{2} - 10036 x^{3} + 37249 x^{4}$ | |
| Frobenius angles: | $\pm0.0758389534121$, $\pm0.143734387197$ |
| Angle rank: | $2$ (numerical) |
| Jacobians: | $7$ |
This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
| $p$-rank: | $2$ |
| Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $28223$ | $1365964977$ | $51645176483264$ | $1925087193828353529$ | $71708977964993290916423$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $142$ | $36668$ | $7183858$ | $1387462228$ | $267785457142$ | $51682551580478$ | $9974730545058022$ | $1925122956304959076$ | $371548729957560657394$ | $71708904873772233393068$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 7 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=155x^6+57x^5+130x^4+60x^3+130x^2+57x+155$
- $y^2=14x^6+167x^5+27x^4+8x^3+27x^2+167x+14$
- $y^2=181x^6+157x^5+78x^4+189x^3+78x^2+157x+181$
- $y^2=171x^6+87x^5+111x^4+83x^3+111x^2+87x+171$
- $y^2=89x^6+184x^5+176x^4+119x^3+176x^2+184x+89$
- $y^2=44x^6+5x^5+123x^4+89x^3+123x^2+5x+44$
- $y^2=115x^6+100x^5+135x^4+143x^3+135x^2+100x+115$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{193}$.
Endomorphism algebra over $\F_{193}$| The isogeny class factors as 1.193.abb $\times$ 1.193.az and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
Base change
This is a primitive isogeny class.