Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 6 x + 18 x^{2} + 78 x^{3} + 169 x^{4}$ |
| Frobenius angles: | $\pm0.450221630168$, $\pm0.950221630168$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(i, \sqrt{17})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $8$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $272$ | $28288$ | $5111696$ | $800210944$ | $137860719632$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $20$ | $170$ | $2324$ | $28014$ | $371300$ | $4826810$ | $62769860$ | $815694814$ | $10604329652$ | $137858491850$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=3 x^6+7 x^5+8 x^4+4 x^3+12 x^2+4 x+1$
- $y^2=11 x^6+5 x^5+10 x^4+8 x^3+9 x$
- $y^2=12 x^6+12 x^5+8 x^4+8 x^2+x+12$
- $y^2=4 x^6+12 x^5+x^4+x^2+x+4$
- $y^2=8 x^6+2 x^5+4 x^4+12 x^3+3 x^2+8 x+3$
- $y^2=x^6+2 x^5+x^4+12 x^3+9 x^2+12 x+3$
- $y^2=11 x^6+8 x^5+6 x^4+10 x^3+5 x^2+2 x+1$
- $y^2=9 x^6+9 x^4+2 x^3+12 x^2+9 x+8$
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 \(\Q(i, \sqrt{17})\). |
| The base change of $A$ to $\F_{13^{4}}$ is 1.28561.ako 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-17}) \)$)$ |
- Endomorphism algebra over $\F_{13^{2}}$
The base change of $A$ to $\F_{13^{2}}$ is the simple isogeny class 2.169.a_ako and its endomorphism algebra is \(\Q(i, \sqrt{17})\).
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.ag_s | $2$ | 2.169.a_ako |
| 2.13.a_ai | $8$ | (not in LMFDB) |
| 2.13.a_i | $8$ | (not in LMFDB) |