Invariants
| Base field: | $\F_{31}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - x + 31 x^{2} )^{2}$ |
| $1 - 2 x + 63 x^{2} - 62 x^{3} + 961 x^{4}$ | |
| Frobenius angles: | $\pm0.471376367478$, $\pm0.471376367478$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $12$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $31$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $961$ | $1046529$ | $893053456$ | $849573288729$ | $819362057499001$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $30$ | $1084$ | $29976$ | $919924$ | $28619850$ | $887605918$ | $27513004710$ | $852888258724$ | $26439607273416$ | $819628358233804$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):
- $y^2=6 x^6+3 x^5+16 x^4+9 x^3+7 x^2+27 x+13$
- $y^2=10 x^6+26 x^5+2 x^4+20 x^3+2 x^2+26 x+10$
- $y^2=30 x^6+6 x^5+12 x^4+25 x^3+24 x^2+7 x+12$
- $y^2=21 x^6+5 x^5+5 x^4+26 x^3+14 x^2+3 x+24$
- $y^2=16 x^6+6 x^5+20 x^4+3 x^3+20 x^2+6 x+16$
- $y^2=22 x^6+12 x^5+28 x^4+5 x^3+17 x^2+17 x+25$
- $y^2=19 x^6+30 x^5+18 x^4+28 x^3+25 x+3$
- $y^2=6 x^6+4 x^5+4 x^4+24 x^3+4 x^2+13 x+13$
- $y^2=3 x^6+3 x^3+12$
- $y^2=6 x^6+29 x^5+15 x^4+29 x^3+15 x^2+29 x+6$
- $y^2=16 x^6+15 x^5+5 x^4+17 x^3+30 x^2+3 x+22$
- $y^2=28 x^6+27 x^5+3 x^4+10 x^3+3 x^2+27 x+28$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{31}$.
Endomorphism algebra over $\F_{31}$| The isogeny class factors as 1.31.ab 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-123}) \)$)$ |
Base change
This is a primitive isogeny class.