Invariants
Base field: | $\F_{7}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 3 x + 15 x^{2} - 21 x^{3} + 49 x^{4}$ |
Frobenius angles: | $\pm0.335255265575$, $\pm0.477002866158$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.14725.1 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $41$ | $3649$ | $134111$ | $5652301$ | $278689136$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $5$ | $71$ | $389$ | $2355$ | $16580$ | $117587$ | $823583$ | $5763379$ | $40358603$ | $282515486$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=3x^6+x^4+4x^3+5x^2+4x+4$
- $y^2=3x^6+2x^5+3x^4+3x^3+4x^2+x$
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 4.0.14725.1. |
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.d_p | $2$ | 2.49.v_hp |