Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 11 x^{2} + 169 x^{4}$ |
| Frobenius angles: | $\pm0.180475001423$, $\pm0.819524998577$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-15}, \sqrt{37})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $6$ |
| Cyclic group of points: | yes |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $159$ | $25281$ | $4831056$ | $828230841$ | $137857884639$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $14$ | $148$ | $2198$ | $28996$ | $371294$ | $4835302$ | $62748518$ | $815750788$ | $10604499374$ | $137857277428$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=2 x^6+6 x^5+4 x^3+4 x^2+12 x+8$
- $y^2=4 x^6+12 x^5+8 x^3+8 x^2+11 x+3$
- $y^2=9 x^6+3 x^5+x^4+2 x^3+3 x^2+12 x+2$
- $y^2=5 x^6+6 x^5+2 x^4+4 x^3+6 x^2+11 x+4$
- $y^2=3 x^6+10 x^5+3 x^4+5 x^3+9 x^2+9 x+8$
- $y^2=12 x^6+7 x^5+9 x^4+6 x^3+11 x^2+8 x+5$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{13^{2}}$.
Endomorphism algebra over $\F_{13}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-15}, \sqrt{37})\). |
| The base change of $A$ to $\F_{13^{2}}$ is 1.169.al 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-555}) \)$)$ |
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 |
|---|---|---|
| 2.13.a_l | $4$ | (not in LMFDB) |