Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 7 x + 29 x^{2} - 91 x^{3} + 169 x^{4}$ |
| Frobenius angles: | $\pm0.138271059594$, $\pm0.479742145051$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.72557.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $9$ |
| Isomorphism classes: | 9 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $101$ | $29997$ | $4815377$ | $808029189$ | $137973330176$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $7$ | $179$ | $2191$ | $28291$ | $371602$ | $4834667$ | $62771107$ | $815734819$ | $10604535907$ | $137859356534$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 9 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=2x^6+6x^5+2x^4+9x^3+7x+12$
- $y^2=9x^6+8x^5+10x^3+x^2+7x+2$
- $y^2=8x^6+6x^5+3x^4+11x^3+8x^2+6x+5$
- $y^2=8x^6+9x^5+6x^4+8x^3+10x^2+3x+1$
- $y^2=2x^6+2x^5+9x^3+11$
- $y^2=5x^5+12x^4+10x^2+12x+6$
- $y^2=8x^6+3x^5+5x^4+7x^3+8x^2+9x+7$
- $y^2=11x^6+2x^5+9x^4+4x^3+7x^2+7x+1$
- $y^2=12x^5+12x^3+12x^2+1$
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 4.0.72557.1. |
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_bd | $2$ | 2.169.j_adr |