Invariants
Base field: | $\F_{13}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 6 x + 13 x^{2} )( 1 - 3 x + 13 x^{2} )$ |
$1 - 9 x + 44 x^{2} - 117 x^{3} + 169 x^{4}$ | |
Frobenius angles: | $\pm0.187167041811$, $\pm0.363422825076$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $3$ |
Isomorphism classes: | 8 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $88$ | $29920$ | $5070208$ | $823996800$ | $137921504248$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $5$ | $177$ | $2306$ | $28849$ | $371465$ | $4827174$ | $62759597$ | $815785921$ | $10604522378$ | $137857504857$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=5x^6+6x^5+8x^4+6x^3+4x^2+8x+8$
- $y^2=5x^6+6x^5+7x^4+11x^3+12x^2+9x+8$
- $y^2=6x^5+4x^3+7x^2+3x$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{13}$.
Endomorphism algebra over $\F_{13}$The isogeny class factors as 1.13.ag $\times$ 1.13.ad 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.