Invariants
| Base field: | $\F_{71}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 11 x + 71 x^{2} )^{2}$ |
| $1 - 22 x + 263 x^{2} - 1562 x^{3} + 5041 x^{4}$ | |
| Frobenius angles: | $\pm0.273623649113$, $\pm0.273623649113$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $7$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $61$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $3721$ | $25633969$ | $128826437776$ | $646243663070329$ | $3255366961467129601$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $50$ | $5084$ | $359936$ | $25430964$ | $1804297750$ | $128099667278$ | $9095108519050$ | $645753446994724$ | $45848500618080896$ | $3255243555887655404$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 7 curves (of which all are hyperelliptic):
- $y^2=7 x^6+44 x^5+51 x^4+47 x^3+64 x^2+64 x+63$
- $y^2=69 x^6+29 x^5+59 x^4+55 x^3+25 x^2+66 x+42$
- $y^2=25 x^6+56 x^5+33 x^4+40 x^3+13 x^2+51 x+60$
- $y^2=34 x^6+5 x^5+27 x^4+15 x^3+27 x^2+5 x+34$
- $y^2=15 x^6+62 x^5+3 x^4+63 x^3+17 x^2+42 x+47$
- $y^2=29 x^6+50 x^5+70 x^4+53 x^3+2 x^2+40 x+13$
- $y^2=7 x^6+70 x^5+56 x^4+47 x^3+66 x^2+32 x+61$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{71}$.
Endomorphism algebra over $\F_{71}$| The isogeny class factors as 1.71.al 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-163}) \)$)$ |
Base change
This is a primitive isogeny class.