Invariants
| Base field: | $\F_{31}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 54 x^{2} + 961 x^{4}$ |
| Frobenius angles: | $\pm0.0817464088246$, $\pm0.918253591175$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-2}, \sqrt{29})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $4$ |
| Isomorphism classes: | 10 |
This isogeny class is simple but not geometrically 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $908$ | $824464$ | $887501900$ | $851057910784$ | $819628335079628$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $32$ | $854$ | $29792$ | $921534$ | $28629152$ | $887500118$ | $27512614112$ | $852892755454$ | $26439622160672$ | $819628383178454$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which all are hyperelliptic):
- $y^2=23 x^6+12 x^4+9 x^3+9 x^2+16$
- $y^2=25 x^6+24 x^5+6 x^4+23 x^3+5 x^2+27 x+6$
- $y^2=11 x^6+7 x^5+29 x^4+3 x^3+5 x^2+5 x+18$
- $y^2=27 x^6+25 x^5+20 x^4+6 x^3+29 x^2+8 x+16$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{31^{2}}$.
Endomorphism algebra over $\F_{31}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-2}, \sqrt{29})\). |
| The base change of $A$ to $\F_{31^{2}}$ is 1.961.acc 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-58}) \)$)$ |
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
| Twist | Extension degree | Common base change |
|---|---|---|
| 2.31.a_cc | $4$ | (not in LMFDB) |
| 2.31.ae_i | $8$ | (not in LMFDB) |
| 2.31.e_i | $8$ | (not in LMFDB) |