Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 7 x + 30 x^{2} - 91 x^{3} + 169 x^{4}$ |
| Frobenius angles: | $\pm0.155060670890$, $\pm0.472256355230$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.640332.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $4$ |
| Isomorphism classes: | 4 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $102$ | $30396$ | $4861728$ | $810235776$ | $138012101742$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $7$ | $181$ | $2212$ | $28369$ | $371707$ | $4834906$ | $62772871$ | $815728513$ | $10604420260$ | $137858852221$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=6x^6+3x^5+10x^4+4x^3+3x^2+x+10$
- $y^2=6x^6+4x^5+6x^4+12x$
- $y^2=5x^6+7x^5+3x^4+x^3+8x^2+10x$
- $y^2=8x^6+9x^5+11x^4+8x^3+11x^2+10x+6$
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 4.0.640332.1. |
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_be | $2$ | 2.169.l_abk |