Invariants
Base field: | $\F_{7}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 2 x - 2 x^{2} - 14 x^{3} + 49 x^{4}$ |
Frobenius angles: | $\pm0.0805195460407$, $\pm0.700955651792$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.2312.1 |
Galois group: | $D_{4}$ |
Jacobians: | $4$ |
Isomorphism classes: | 8 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $32$ | $2048$ | $97952$ | $5832704$ | $281130912$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $6$ | $42$ | $282$ | $2430$ | $16726$ | $117066$ | $825642$ | $5765310$ | $40354662$ | $282536362$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=x^6+3x^5+4x^4+3x^2+2x+3$
- $y^2=6x^6+x^5+x^4+2x^3+2x^2+4x+5$
- $y^2=2x^6+x^5+x^4+6x^3+4x^2+2x+1$
- $y^2=3x^6+6x^5+6x^3+x^2+4x+3$
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 4.0.2312.1. |
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.c_ac | $2$ | 2.49.ai_bu |