Invariants
Base field: | $\F_{13}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 7 x + 34 x^{2} - 91 x^{3} + 169 x^{4}$ |
Frobenius angles: | $\pm0.219632524221$, $\pm0.436075769158$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.304028.1 |
Galois group: | $D_{4}$ |
Jacobians: | $4$ |
Isomorphism classes: | 4 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $106$ | $32012$ | $5048992$ | $817970624$ | $137907422626$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $7$ | $189$ | $2296$ | $28641$ | $371427$ | $4830762$ | $62762287$ | $815691489$ | $10604093752$ | $137857571509$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=11x^6+5x^5+2x^4+11x^3+5x^2+11x+7$
- $y^2=7x^6+12x^5+x^4+x^3+6x+5$
- $y^2=2x^6+6x^5+10x^4+4x^3+2x^2+12x+10$
- $y^2=11x^6+9x^5+2x^4+10x^3+8x^2+10x+8$
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 4.0.304028.1. |
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.h_bi | $2$ | 2.169.t_im |