Invariants
| Base field: | $\F_{31}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 8 x + 31 x^{2} )^{2}$ |
| $1 - 16 x + 126 x^{2} - 496 x^{3} + 961 x^{4}$ | |
| Frobenius angles: | $\pm0.244865078763$, $\pm0.244865078763$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $14$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $576$ | $921600$ | $901440576$ | $856439193600$ | $820095180388416$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $16$ | $958$ | $30256$ | $927358$ | $28645456$ | $887515198$ | $27512200816$ | $852887374078$ | $26439605665936$ | $819628268587198$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 14 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=x^6+14x^3+16$
- $y^2=x^6+18x^3+4$
- $y^2=16x^6+27x^5+6x^4+7x^3+6x^2+27x+16$
- $y^2=15x^6+11x^4+11x^2+15$
- $y^2=15x^6+17x^5+7x^4+6x^3+7x^2+17x+15$
- $y^2=29x^5+20x^4+25x^3+5x^2+27x$
- $y^2=29x^6+6x^5+10x^4+6x^3+5x^2+17x+23$
- $y^2=10x^6+20x^4+20x^2+10$
- $y^2=22x^6+22x^5+29x^4+29x^2+22x+22$
- $y^2=27x^6+13x^4+13x^2+27$
- $y^2=17x^6+28x^5+16x^4+x^3+16x^2+28x+17$
- $y^2=4x^6+19x^5+9x^4+11x^3+9x^2+19x+4$
- $y^2=6x^6+9x^4+9x^2+6$
- $y^2=30x^6+15x^5+x^4+13x^3+x^2+15x+30$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{31}$.
Endomorphism algebra over $\F_{31}$| The isogeny class factors as 1.31.ai 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$ |
Base change
This is a primitive isogeny class.