Invariants
Base field: | $\F_{13}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 9 x + 45 x^{2} - 117 x^{3} + 169 x^{4}$ |
Frobenius angles: | $\pm0.215685987913$, $\pm0.344616475996$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.20725.1 |
Galois group: | $D_{4}$ |
Jacobians: | $3$ |
Isomorphism classes: | 3 |
This isogeny class is simple and geometrically 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $89$ | $30349$ | $5131829$ | $828193861$ | $138005338064$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $5$ | $179$ | $2333$ | $28995$ | $371690$ | $4825163$ | $62743721$ | $815734819$ | $10604492489$ | $137857962614$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=4x^5+6x^4+12x^3+12x^2+11x+2$
- $y^2=5x^6+8x^5+12x^3+8x^2+11x+7$
- $y^2=2x^6+10x^5+9x^4+8x^3+10x+6$
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 4.0.20725.1. |
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_bt | $2$ | 2.169.j_jx |