Invariants
Base field: | $\F_{13}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 6 x + 13 x^{2} )( 1 - x + 13 x^{2} )$ |
$1 - 7 x + 32 x^{2} - 91 x^{3} + 169 x^{4}$ | |
Frobenius angles: | $\pm0.187167041811$, $\pm0.455715642762$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $6$ |
Isomorphism classes: | 40 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $104$ | $31200$ | $4954976$ | $814320000$ | $138011646344$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $7$ | $185$ | $2254$ | $28513$ | $371707$ | $4833830$ | $62770519$ | $815705953$ | $10604190982$ | $137857941425$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=8x^6+5x^5+3x^4+7x^3+x^2+11x+6$
- $y^2=11x^6+3x^5+4x^4+7x^3+7x+7$
- $y^2=2x^6+9x^5+3x^4+5x^3+12x^2+3x$
- $y^2=6x^6+8x^5+x^4+2x^3+9x^2+8x+2$
- $y^2=x^6+11x^5+2x^4+x^3+9x^2+11x+11$
- $y^2=x^5+4x^4+9x^3+4x^2+2x+2$
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.ab 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.