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: | $14$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2, 3$ |
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 is principally polarizable and contains the Jacobians of 14 curves (of which all are hyperelliptic):
- $y^2=x^6+14 x^3+16$
- $y^2=x^6+18 x^3+4$
- $y^2=16 x^6+27 x^5+6 x^4+7 x^3+6 x^2+27 x+16$
- $y^2=15 x^6+11 x^4+11 x^2+15$
- $y^2=15 x^6+17 x^5+7 x^4+6 x^3+7 x^2+17 x+15$
- $y^2=29 x^5+20 x^4+25 x^3+5 x^2+27 x$
- $y^2=29 x^6+6 x^5+10 x^4+6 x^3+5 x^2+17 x+23$
- $y^2=10 x^6+20 x^4+20 x^2+10$
- $y^2=22 x^6+22 x^5+29 x^4+29 x^2+22 x+22$
- $y^2=27 x^6+13 x^4+13 x^2+27$
- $y^2=17 x^6+28 x^5+16 x^4+x^3+16 x^2+28 x+17$
- $y^2=4 x^6+19 x^5+9 x^4+11 x^3+9 x^2+19 x+4$
- $y^2=6 x^6+9 x^4+9 x^2+6$
- $y^2=30 x^6+15 x^5+x^4+13 x^3+x^2+15 x+30$
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.