Invariants
Base field: | $\F_{13}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 7 x + 13 x^{2} )( 1 + x + 13 x^{2} )$ |
$1 - 6 x + 19 x^{2} - 78 x^{3} + 169 x^{4}$ | |
Frobenius angles: | $\pm0.0772104791556$, $\pm0.544284357238$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $8$ |
Isomorphism classes: | 24 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $105$ | $28665$ | $4596480$ | $802190025$ | $137990339025$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $8$ | $172$ | $2090$ | $28084$ | $371648$ | $4829254$ | $62737424$ | $815726116$ | $10604813330$ | $137859184732$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=7x^6+3x^5+3x+7$
- $y^2=2x^6+11x^5+x^4+3x^3+6x^2+5x+2$
- $y^2=8x^6+2x^5+7x^4+9x^3+4x^2+12x+9$
- $y^2=8x^6+5x^5+10x^4+9x^3+4x^2+11$
- $y^2=2x^6+4x^5+8x^4+x^3+11x^2+5x+11$
- $y^2=5x^6+x^5+9x^4+9x^2+3x+1$
- $y^2=6x^6+3x^5+12x^4+11x^3+12x^2+6x+7$
- $y^2=2x^6+9x^5+6x^4+5x^3+8x^2+3x+11$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{13}$.
Endomorphism algebra over $\F_{13}$The isogeny class factors as 1.13.ah $\times$ 1.13.b 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.