Invariants
Base field: | $\F_{13}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 9 x + 42 x^{2} - 117 x^{3} + 169 x^{4}$ |
Frobenius angles: | $\pm0.136139978944$, $\pm0.390198274089$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.119068.1 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $86$ | $29068$ | $4947752$ | $815299264$ | $137654604926$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $5$ | $173$ | $2252$ | $28545$ | $370745$ | $4828394$ | $62774717$ | $815838625$ | $10604648252$ | $137858097413$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=4x^6+x^5+9x^3+3x^2+8x$
- $y^2=5x^6+10x^5+6x^4+10x^2+10x+7$
- $y^2=3x^5+x^4+7x^3+7x^2+11x+8$
- $y^2=8x^6+10x^5+7x^4+7x^3+9x^2+6$
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 4.0.119068.1. |
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_bq | $2$ | 2.169.d_ae |