Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 8 x + 32 x^{2} - 104 x^{3} + 169 x^{4}$ |
| Frobenius angles: | $\pm0.0370621216586$, $\pm0.462937878341$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(i, \sqrt{10})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $3$ |
| Isomorphism classes: | 6 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $90$ | $28260$ | $4705290$ | $798627600$ | $137233172250$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $6$ | $170$ | $2142$ | $27958$ | $369606$ | $4826810$ | $62749182$ | $815662558$ | $10604218086$ | $137858491850$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=11x^6+x^5+10x^4+2x^3+6x^2+11x+6$
- $y^2=5x^6+5x^5+6x^4+2x^2+10x+6$
- $y^2=6x^6+12x^5+4x^4+4x^2+x+6$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{13^{4}}$.
Endomorphism algebra over $\F_{13}$| The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{10})\). |
| The base change of $A$ to $\F_{13^{4}}$ is 1.28561.alq 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-10}) \)$)$ |
- Endomorphism algebra over $\F_{13^{2}}$
The base change of $A$ to $\F_{13^{2}}$ is the simple isogeny class 2.169.a_alq and its endomorphism algebra is \(\Q(i, \sqrt{10})\).
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.i_bg | $2$ | 2.169.a_alq |
| 2.13.a_ag | $8$ | (not in LMFDB) |
| 2.13.a_g | $8$ | (not in LMFDB) |