Invariants
Base field: | $\F_{7}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 5 x + 7 x^{2} )( 1 + 2 x + 7 x^{2} )$ |
$1 - 3 x + 4 x^{2} - 21 x^{3} + 49 x^{4}$ | |
Frobenius angles: | $\pm0.106147807505$, $\pm0.623375857214$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $2$ |
Isomorphism classes: | 18 |
This isogeny class is not 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $30$ | $2340$ | $100440$ | $5709600$ | $287002650$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $5$ | $49$ | $290$ | $2377$ | $17075$ | $117466$ | $824045$ | $5773873$ | $40361870$ | $282483289$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=3x^6+4x^5+2x^4+x^3+5x+5$
- $y^2=4x^6+5x^5+6x^4+3x^3+5x^2+6x+2$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{7}$.
Endomorphism algebra over $\F_{7}$The isogeny class factors as 1.7.af $\times$ 1.7.c and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
Base change
This is a primitive isogeny class.