Invariants
| Base field: | $\F_{7^{2}}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 12 x + 49 x^{2} )^{2}$ |
| $1 - 24 x + 242 x^{2} - 1176 x^{3} + 2401 x^{4}$ | |
| Frobenius angles: | $\pm0.172237328522$, $\pm0.172237328522$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $5$ |
This isogeny class is not simple, not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1444$ | $5550736$ | $13849994596$ | $33263917830144$ | $79809480702695524$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $26$ | $2310$ | $117722$ | $5770174$ | $282536186$ | $13841755206$ | $678225703034$ | $33232939199614$ | $1628413572591578$ | $79792265570914950$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 5 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=ax^5+ax^4+4ax^3+6ax+3a$
- $y^2=6ax^6+(a+5)x^5+(5a+5)x^4+(a+6)x^3+2ax^2+4x$
- $y^2=(5a+6)x^6+(5a+6)x^5+5ax^4+(5a+3)x^3+5ax^2+(5a+6)x+5a+6$
- $y^2=ax^6+(3a+1)x^5+4x^4+(6a+3)x^3+4x^2+(3a+1)x+a$
- $y^2=6x^6+(a+5)x^4+(a+5)x^2+6$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{7^{2}}$.
Endomorphism algebra over $\F_{7^{2}}$| The isogeny class factors as 1.49.am 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-13}) \)$)$ |
Base change
This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{7^{2}}$.
| Subfield | Primitive Model |
| $\F_{7}$ | 2.7.a_am |