Invariants
Base field: | $\F_{7}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 4 x + 11 x^{2} - 28 x^{3} + 49 x^{4}$ |
Frobenius angles: | $\pm0.158901191781$, $\pm0.538942184569$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.138768.1 |
Galois group: | $D_{4}$ |
Jacobians: | $2$ |
Isomorphism classes: | 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $29$ | $2697$ | $112868$ | $5655609$ | $288523349$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $4$ | $56$ | $328$ | $2356$ | $17164$ | $118838$ | $823876$ | $5765284$ | $40367704$ | $282477416$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=x^6+4x^5+4x^2+2x+6$
- $y^2=4x^6+4x^5+2x^4+2x+3$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{7}$.
Endomorphism algebra over $\F_{7}$The endomorphism algebra of this simple isogeny class is 4.0.138768.1. |
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 |
---|---|---|
2.7.e_l | $2$ | 2.49.g_af |