Invariants
| Base field: | $\F_{71}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 138 x^{2} + 5041 x^{4}$ |
| Frobenius angles: | $\pm0.0378656773237$, $\pm0.962134322676$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(i, \sqrt{70})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $6$ |
| Isomorphism classes: | 44 |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
This isogeny class is simple but not geometrically 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $4904$ | $24049216$ | $128099742824$ | $645298183398400$ | $3255243549667372904$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $72$ | $4766$ | $357912$ | $25393758$ | $1804229352$ | $128099201726$ | $9095120158392$ | $645753472257598$ | $45848500718449032$ | $3255243548324864606$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=21 x^6+56 x^5+18 x^4+35 x^3+12 x^2+61 x+46$
- $y^2=68 x^6+30 x^5+43 x^4+65 x^2+57 x+6$
- $y^2=47 x^6+34 x^5+10 x^4+46 x^3+65 x^2+65 x+19$
- $y^2=x^6+11 x^4+38 x^3+11 x^2+58 x+38$
- $y^2=9 x^6+53 x^5+13 x^4+53 x^2+3 x+58$
- $y^2=62 x^6+28 x^5+23 x^4+14 x^2+6 x+66$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{71^{2}}$.
Endomorphism algebra over $\F_{71}$| The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{70})\). |
| The base change of $A$ to $\F_{71^{2}}$ is 1.5041.afi 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-70}) \)$)$ |
Base change
This is a primitive isogeny class.