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: | $\pm0.755134921237$, $\pm0.755134921237$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $14$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2, 5$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1600$ | $921600$ | $873793600$ | $856439193600$ | $819161641000000$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $48$ | $958$ | $29328$ | $927358$ | $28612848$ | $887515198$ | $27513027408$ | $852887374078$ | $26439638655408$ | $819628268587198$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 14 curves (of which all are hyperelliptic):
- $y^2=3 x^6+11 x^3+17$
- $y^2=3 x^6+23 x^3+12$
- $y^2=26 x^6+9 x^5+2 x^4+23 x^3+2 x^2+9 x+26$
- $y^2=5 x^6+14 x^4+14 x^2+5$
- $y^2=5 x^6+16 x^5+23 x^4+2 x^3+23 x^2+16 x+5$
- $y^2=20 x^5+17 x^4+29 x^3+12 x^2+9 x$
- $y^2=20 x^6+2 x^5+24 x^4+2 x^3+12 x^2+16 x+18$
- $y^2=30 x^6+29 x^4+29 x^2+30$
- $y^2=28 x^6+28 x^5+20 x^4+20 x^2+28 x+28$
- $y^2=9 x^6+25 x^4+25 x^2+9$
- $y^2=16 x^6+30 x^5+26 x^4+21 x^3+26 x^2+30 x+16$
- $y^2=22 x^6+27 x^5+3 x^4+14 x^3+3 x^2+27 x+22$
- $y^2=18 x^6+27 x^4+27 x^2+18$
- $y^2=10 x^6+5 x^5+21 x^4+25 x^3+21 x^2+5 x+10$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{31}$.
Endomorphism algebra over $\F_{31}$| The isogeny class factors as 1.31.i 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$ |
Base change
This is a primitive isogeny class.