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: | $6$ |
| Isomorphism classes: | 6 |
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 contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=12x^6+7x^5+8x^4+4x^3+7x^2+7x+2$
- $y^2=9x^6+12x^5+2x^4+6x^3+5x^2+x+11$
- $y^2=9x^5+8x^4+11x^3+x^2+2x+6$
- $y^2=12x^6+11x^5+3x^4+11x^3+4x^2+4x+1$
- $y^2=5x^6+12x^5+12x^3+6x^2+9x+10$
- $y^2=7x^6+5x^5+9x^4+9x^3+10x^2+3x+11$
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 |