Invariants
| Base field: | $\F_{31}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 58 x^{2} + 961 x^{4}$ |
| Frobenius angles: | $\pm0.0574820589755$, $\pm0.942517941024$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(i, \sqrt{30})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $6$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $904$ | $817216$ | $887475784$ | $850231526400$ | $819628300316104$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $32$ | $846$ | $29792$ | $920638$ | $28629152$ | $887447886$ | $27512614112$ | $852890572798$ | $26439622160672$ | $819628313651406$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=28 x^6+30 x^5+28 x^4+16 x^3+7 x^2+15$
- $y^2=15 x^6+15 x^5+2 x^4+10 x^3+19 x^2+16 x+13$
- $y^2=30 x^5+12 x^4+x^3+9 x^2+11 x+11$
- $y^2=7 x^6+4 x^5+3 x^4+26 x^3+15 x^2+19 x+4$
- $y^2=19 x^6+17 x^5+11 x^4+15 x^2+4 x+7$
- $y^2=16 x^6+27 x^5+26 x^4+13 x^2+x+2$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{31^{2}}$.
Endomorphism algebra over $\F_{31}$| The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{30})\). |
| The base change of $A$ to $\F_{31^{2}}$ is 1.961.acg 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-30}) \)$)$ |
Base change
This is a primitive isogeny class.