Invariants
Base field: | $\F_{7}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 4 x + 16 x^{2} - 28 x^{3} + 49 x^{4}$ |
Frobenius angles: | $\pm0.276763328290$, $\pm0.464689692576$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.28928.2 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $34$ | $3332$ | $133858$ | $5771024$ | $281756674$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $4$ | $66$ | $388$ | $2406$ | $16764$ | $117858$ | $823036$ | $5758014$ | $40342564$ | $282515266$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=2x^6+5x^5+3x^4+3x^3+2x^2+2x+2$
- $y^2=6x^6+3x^5+2x^4+3x^3+6x^2+5x+3$
- $y^2=5x^6+5x^5+2x^4+6x^3+6x^2+4x+5$
- $y^2=5x^6+6x^5+4x^4+3x^3+x^2+x+3$
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 4.0.28928.2. |
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.e_q | $2$ | 2.49.q_fa |