Invariants
| Base field: | $\F_{31}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 + 3 x + 31 x^{2} )^{2}$ |
| $1 + 6 x + 71 x^{2} + 186 x^{3} + 961 x^{4}$ | |
| Frobenius angles: | $\pm0.586827998080$, $\pm0.586827998080$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $11$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $5, 7$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1225$ | $1030225$ | $872611600$ | $851255343225$ | $820228120140625$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $38$ | $1068$ | $29288$ | $921748$ | $28650098$ | $887495838$ | $27511988318$ | $852893157988$ | $26439635198648$ | $819628182129948$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 11 curves (of which all are hyperelliptic):
- $y^2=5 x^6+13 x^5+23 x^4+30 x^3+23 x^2+17 x+28$
- $y^2=x^6+27 x^5+23 x^4+28 x^3+23 x^2+27 x+1$
- $y^2=16 x^6+x^5+7 x^4+19 x^3+21 x^2+19$
- $y^2=2 x^6+15 x^5+11 x^3+17 x^2+18 x+22$
- $y^2=22 x^6+27 x^5+19 x^4+29 x^3+16 x^2+14 x+23$
- $y^2=23 x^6+24 x^5+11 x^4+30 x^3+7 x^2+20 x+14$
- $y^2=8 x^6+18 x^5+23 x^4+28 x^3+23 x^2+18 x+8$
- $y^2=26 x^6+2 x^5+29 x^3+2 x^2+16 x+8$
- $y^2=9 x^6+27 x^4+27 x^3+27 x^2+9$
- $y^2=27 x^6+9 x^5+4 x^4+6 x^3+4 x^2+9 x+27$
- $y^2=2 x^6+6 x^5+30 x^4+25 x^3+30 x^2+6 x+2$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{31}$.
Endomorphism algebra over $\F_{31}$| The isogeny class factors as 1.31.d 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-115}) \)$)$ |
Base change
This is a primitive isogeny class.