Invariants
Base field: | $\F_{13}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 10 x + 49 x^{2} - 130 x^{3} + 169 x^{4}$ |
Frobenius angles: | $\pm0.151058869957$, $\pm0.334339837461$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.27200.2 |
Galois group: | $D_{4}$ |
Jacobians: | $2$ |
Isomorphism classes: | 2 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $79$ | $28361$ | $5005756$ | $823631801$ | $137951509639$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $4$ | $168$ | $2278$ | $28836$ | $371544$ | $4826622$ | $62756488$ | $815805828$ | $10604793214$ | $137858817928$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=11x^6+4x^5+x^4+12x^3+9x^2+12x+5$
- $y^2=x^6+9x^5+12x^4+12x^3+9x^2+6x+11$
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 4.0.27200.2. |
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.k_bx | $2$ | 2.169.ac_fj |