Invariants
| Base field: | $\F_{7}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 + 7 x^{2} )( 1 + 2 x + 7 x^{2} )$ |
| $1 + 2 x + 14 x^{2} + 14 x^{3} + 49 x^{4}$ | |
| Frobenius angles: | $\pm0.5$, $\pm0.623375857214$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $4$ |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
| $p$-rank: | $1$ |
| Slopes: | $[0, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $80$ | $3840$ | $106640$ | $5529600$ | $286576400$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $10$ | $74$ | $310$ | $2302$ | $17050$ | $117866$ | $822790$ | $5764798$ | $40349290$ | $282483914$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which all are hyperelliptic):
- $y^2=2 x^6+2 x^5+x^4+5 x^3+x^2+2 x+2$
- $y^2=4 x^6+x^5+6 x^4+2 x^3+6 x^2+x+4$
- $y^2=x^6+2 x^5+2 x+1$
- $y^2=6 x^6+x^5+4 x^4+x^3+4 x^2+x+6$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{7^{2}}$.
Endomorphism algebra over $\F_{7}$| The isogeny class factors as 1.7.a $\times$ 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: |
The base change of $A$ to $\F_{7^{2}}$ is 1.49.k $\times$ 1.49.o. The endomorphism algebra for each factor is:
|
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.ac_o | $2$ | 2.49.y_je |