Invariants
Base field: | $\F_{13}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 - 6 x + 13 x^{2} )( 1 - 9 x + 42 x^{2} - 117 x^{3} + 169 x^{4} )$ |
$1 - 15 x + 109 x^{2} - 486 x^{3} + 1417 x^{4} - 2535 x^{5} + 2197 x^{6}$ | |
Frobenius angles: | $\pm0.136139978944$, $\pm0.187167041811$, $\pm0.390198274089$ |
Angle rank: | $3$ (numerical) |
This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.
Newton polygon
This isogeny class is ordinary.
$p$-rank: | $3$ |
Slopes: | $[0, 0, 0, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $688$ | $4650880$ | $10964218432$ | $23480618803200$ | $51274688479675888$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-1$ | $163$ | $2270$ | $28783$ | $371939$ | $4832464$ | $62783615$ | $815839103$ | $10604535446$ | $137857414363$ |
Jacobians and polarizations
This isogeny class is principally polarizable, but does not contain a Jacobian.
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 $\times$ 2.13.aj_bq 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.