Invariants
| Base field: | $\F_{37}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 4 x - 21 x^{2} - 148 x^{3} + 1369 x^{4}$ |
| Frobenius angles: | $\pm0.0600231462171$, $\pm0.726689812884$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}, \sqrt{11})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $6$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $3$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1197$ | $1796697$ | $2527475076$ | $3513651197049$ | $4807435964119677$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $34$ | $1312$ | $49894$ | $1874788$ | $69327394$ | $2565640222$ | $94932317626$ | $3512476097476$ | $129961745539918$ | $4808584508096032$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=31 x^6+22 x^5+30 x^4+22 x^3+2 x^2+11 x+5$
- $y^2=x^6+2 x^3+3$
- $y^2=x^6+x^3+30$
- $y^2=4 x^6+29 x^5+2 x^4+2 x^3+28 x^2+8 x+24$
- $y^2=29 x^6+13 x^5+5 x^4+3 x^3+25 x^2+27 x+35$
- $y^2=33 x^6+15 x^5+6 x^4+4 x^3+27 x^2+33 x+7$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{37^{3}}$.
Endomorphism algebra over $\F_{37}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{11})\). |
| The base change of $A$ to $\F_{37^{3}}$ is 1.50653.aoq 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-33}) \)$)$ |
Base change
This is a primitive isogeny class.