Invariants
Base field: | $\F_{13}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 5 x + 13 x^{2} )( 1 - 3 x + 13 x^{2} )$ |
$1 - 8 x + 41 x^{2} - 104 x^{3} + 169 x^{4}$ | |
Frobenius angles: | $\pm0.256122854178$, $\pm0.363422825076$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $4$ |
Isomorphism classes: | 6 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $99$ | $31977$ | $5189184$ | $826829289$ | $137766360699$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $6$ | $188$ | $2358$ | $28948$ | $371046$ | $4822598$ | $62738094$ | $815728996$ | $10604516814$ | $137858329868$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=5x^6+4x^5+3x^4+9x^3+3x^2+4x+5$
- $y^2=8x^6+4x^5+8x^4+7x^3+8x^2+4x+8$
- $y^2=x^6+x^5+11x^4+3x^3+11x^2+x+1$
- $y^2=11x^6+3x^5+7x^4+9x^3+7x^2+3x+11$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{13}$.
Endomorphism algebra over $\F_{13}$The isogeny class factors as 1.13.af $\times$ 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.