Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 8 x + 40 x^{2} + 104 x^{3} + 169 x^{4}$ |
| Frobenius angles: | $\pm0.616740354477$, $\pm0.770339669950$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-34 +8 \sqrt{2}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $4$ |
| Cyclic group of points: | yes |
This isogeny class is simple and 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $322$ | $31556$ | $4536658$ | $824116496$ | $137890627042$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $22$ | $186$ | $2062$ | $28854$ | $371382$ | $4825866$ | $62745166$ | $815736798$ | $10604668150$ | $137857406746$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which all are hyperelliptic):
- $y^2=x^6+2 x^5+2 x^4+7 x^3+5 x^2+2$
- $y^2=x^6+9 x^5+7 x^4+7 x^3+4 x^2+9 x+4$
- $y^2=2 x^6+8 x^5+5 x^4+9 x^3+4 x^2+9 x+12$
- $y^2=9 x^6+7 x^5+3 x^4+5 x^3+10 x^2+11 x+3$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{13}$.
Endomorphism algebra over $\F_{13}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-34 +8 \sqrt{2}})\). |
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.ai_bo | $2$ | 2.169.q_ko |