Invariants
Base field: | $\F_{13}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 8 x + 36 x^{2} - 104 x^{3} + 169 x^{4}$ |
Frobenius angles: | $\pm0.147614849952$, $\pm0.431019279425$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.297216.1 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $94$ | $29892$ | $4918174$ | $812225424$ | $137766260974$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $6$ | $178$ | $2238$ | $28438$ | $371046$ | $4832098$ | $62780094$ | $815788126$ | $10604413734$ | $137858130418$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=6x^6+2x^5+11x^4+12x^3+3x+7$
- $y^2=5x^6+2x^5+5x^4+8x^3+7x^2+2x+7$
- $y^2=6x^6+11x^5+7x^4+5x^3+10x^2+12x+3$
- $y^2=3x^5+5x^4+2x^3+10x^2+2x+11$
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 4.0.297216.1. |
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.i_bk | $2$ | 2.169.i_abe |