Invariants
| Base field: | $\F_{7^{2}}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 7 x )^{2}( 1 - 10 x + 49 x^{2} )$ |
| $1 - 24 x + 238 x^{2} - 1176 x^{3} + 2401 x^{4}$ | |
| Frobenius angles: | $0$, $0$, $\pm0.246751714429$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $6$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1440$ | $5529600$ | $13815787680$ | $33232896000000$ | $79789818658327200$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $26$ | $2302$ | $117434$ | $5764798$ | $282466586$ | $13841066302$ | $678220347194$ | $33232907548798$ | $1628413455141146$ | $79792265675109502$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(2a+2)x^6+3x^5+(6a+1)x^4+4x^3+(6a+6)x^2+(2a+6)x+3a+4$
- $y^2=(a+4)x^6+4x^5+(3a+5)x^4+(4a+3)x^3+(4a+1)x^2+3x+a+2$
- $y^2=5ax^6+3ax^5+3ax+5a$
- $y^2=2ax^6+2ax^5+ax^4+5ax^3+ax^2+2ax+2a$
- $y^2=3x^6+(6a+3)x^5+2x^4+(5a+4)x^3+(2a+6)x^2+(a+4)x+4a+5$
- $y^2=(5a+4)x^6+3x^5+ax^4+4x^3+(a+5)x^2+(5a+1)x+4a$
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.ao $\times$ 1.49.ak 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.