Invariants
Base field: | $\F_{13}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 5 x + 13 x^{2} )( 1 - 2 x + 13 x^{2} )$ |
$1 - 7 x + 36 x^{2} - 91 x^{3} + 169 x^{4}$ | |
Frobenius angles: | $\pm0.256122854178$, $\pm0.410543812489$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $7$ |
Isomorphism classes: | 30 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $108$ | $32832$ | $5143824$ | $821193984$ | $137699760348$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $7$ | $193$ | $2338$ | $28753$ | $370867$ | $4825798$ | $62750527$ | $815710081$ | $10604262634$ | $137857931593$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 7 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=2x^6+9x^5+6x^4+4x^3+12x^2+2x+7$
- $y^2=4x^5+9x^3+x^2+7x+11$
- $y^2=9x^6+6x^5+6x^4+10x^3+2x^2+10x+2$
- $y^2=11x^6+6x^5+10x^4+11x^3+x^2+4x+7$
- $y^2=11x^6+3x^5+6x^4+4x^3+8x^2+4x+5$
- $y^2=6x^6+11x^5+7x^4+2x^3+8x^2+8x+8$
- $y^2=6x^6+x^5+8x^3+8x^2+3x+2$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{13^{3}}$.
Endomorphism algebra over $\F_{13}$The isogeny class factors as 1.13.af $\times$ 1.13.ac and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
The base change of $A$ to $\F_{13^{3}}$ is 1.2197.cs 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$ |
Base change
This is a primitive isogeny class.