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: | $\pm0.528623632522$, $\pm0.528623632522$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $12$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $3, 11$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1089$ | $1046529$ | $882090000$ | $849573288729$ | $819894674242809$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $34$ | $1084$ | $29608$ | $919924$ | $28638454$ | $887605918$ | $27512223514$ | $852888258724$ | $26439637047928$ | $819628358233804$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):
- $y^2=2 x^6+x^5+26 x^4+3 x^3+23 x^2+9 x+25$
- $y^2=24 x^6+19 x^5+11 x^4+17 x^3+11 x^2+19 x+24$
- $y^2=28 x^6+18 x^5+5 x^4+13 x^3+10 x^2+21 x+5$
- $y^2=7 x^6+12 x^5+12 x^4+19 x^3+15 x^2+x+8$
- $y^2=17 x^6+18 x^5+29 x^4+9 x^3+29 x^2+18 x+17$
- $y^2=4 x^6+5 x^5+22 x^4+15 x^3+20 x^2+20 x+13$
- $y^2=27 x^6+10 x^5+6 x^4+30 x^3+29 x+1$
- $y^2=2 x^6+22 x^5+22 x^4+8 x^3+22 x^2+25 x+25$
- $y^2=x^6+x^3+4$
- $y^2=18 x^6+25 x^5+14 x^4+25 x^3+14 x^2+25 x+18$
- $y^2=26 x^6+5 x^5+12 x^4+16 x^3+10 x^2+x+28$
- $y^2=22 x^6+19 x^5+9 x^4+30 x^3+9 x^2+19 x+22$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{31}$.
Endomorphism algebra over $\F_{31}$| The isogeny class factors as 1.31.b 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-123}) \)$)$ |
Base change
This is a primitive isogeny class.