Invariants
Base field: | $\F_{13}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 7 x + 27 x^{2} - 91 x^{3} + 169 x^{4}$ |
Frobenius angles: | $\pm0.100578918488$, $\pm0.493559382189$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.6525.1 |
Galois group: | $D_{4}$ |
Jacobians: | $8$ |
Isomorphism classes: | 10 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $99$ | $29205$ | $4723191$ | $803283525$ | $137817648144$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $7$ | $175$ | $2149$ | $28123$ | $371182$ | $4832575$ | $62760229$ | $815721043$ | $10604658967$ | $137859961750$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=7x^6+9x^5+9x^4+12x^3+x^2+8x+7$
- $y^2=9x^6+11x^5+6x^4+10x^3+7x^2+11x+8$
- $y^2=2x^6+4x^5+3x^4+11x^3+6x^2+9x+2$
- $y^2=11x^6+4x^4+10x^3+x^2+11x+5$
- $y^2=2x^5+9x^4+11x^3+9x+2$
- $y^2=x^6+9x^5+10x^4+x^3+3x^2+2x+5$
- $y^2=5x^6+9x^5+x^4+12x^3+4x^2+7x$
- $y^2=7x^6+6x^5+10x^4+11x^3+6x^2+7x+8$
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 4.0.6525.1. |
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_bb | $2$ | 2.169.f_ahz |