Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 7 x + 28 x^{2} - 91 x^{3} + 169 x^{4}$ |
| Frobenius angles: | $\pm0.120377326548$, $\pm0.486822699267$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.154652.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $6$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $100$ | $29600$ | $4769200$ | $805712000$ | $137908520500$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $7$ | $177$ | $2170$ | $28209$ | $371427$ | $4833894$ | $62766991$ | $815733441$ | $10604623330$ | $137859767257$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=12 x^6+7 x^5+8 x^4+4 x^3+7 x^2+7 x+2$
- $y^2=9 x^6+12 x^5+2 x^4+6 x^3+5 x^2+x+11$
- $y^2=9 x^5+8 x^4+11 x^3+x^2+2 x+6$
- $y^2=12 x^6+11 x^5+3 x^4+11 x^3+4 x^2+4 x+1$
- $y^2=5 x^6+12 x^5+12 x^3+6 x^2+9 x+10$
- $y^2=7 x^6+5 x^5+9 x^4+9 x^3+10 x^2+3 x+11$
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 4.0.154652.1. |
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.h_bc | $2$ | 2.169.h_afw |