Invariants
Base field: | $\F_{7}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 3 x + 8 x^{2} - 21 x^{3} + 49 x^{4}$ |
Frobenius angles: | $\pm0.190450914538$, $\pm0.583503963728$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.252648.2 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $34$ | $2788$ | $111928$ | $5821344$ | $291094774$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $5$ | $57$ | $326$ | $2425$ | $17315$ | $118266$ | $822701$ | $5766865$ | $40354634$ | $282413937$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=3x^6+4x^5+5x^4+5x^3+5x+3$
- $y^2=5x^5+4x^4+3x^3+5x^2+2x+5$
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 4.0.252648.2. |
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_i | $2$ | 2.49.h_bk |