Invariants
Base field: | $\F_{13}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 9 x + 39 x^{2} - 117 x^{3} + 169 x^{4}$ |
Frobenius angles: | $\pm0.0228181011636$, $\pm0.419357734967$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.10933.1 |
Galois group: | $D_{4}$ |
Jacobians: | $1$ |
Isomorphism classes: | 1 |
This isogeny class is simple and geometrically simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
$p$-rank: | $1$ |
Slopes: | $[0, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $83$ | $27805$ | $4765943$ | $801479125$ | $137005100048$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $5$ | $167$ | $2171$ | $28059$ | $368990$ | $4823039$ | $62750147$ | $815707891$ | $10604178533$ | $137857322582$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
- $y^2=4x^5+3x^4+11x^3+11x^2+6x+11$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{13}$.
Endomorphism algebra over $\F_{13}$The endomorphism algebra of this simple isogeny class is 4.0.10933.1. |
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
Twist | Extension degree | Common base change |
---|---|---|
2.13.j_bn | $2$ | 2.169.ad_ajn |