Invariants
Base field: | $\F_{13}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 8 x + 34 x^{2} - 104 x^{3} + 169 x^{4}$ |
Frobenius angles: | $\pm0.104164352389$, $\pm0.448054596667$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.1088.2 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $92$ | $29072$ | $4811324$ | $805643264$ | $137559158172$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $6$ | $174$ | $2190$ | $28206$ | $370486$ | $4830942$ | $62773374$ | $815765214$ | $10604496486$ | $137859182734$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=5x^6+4x^5+x^4+4x^3+11x+8$
- $y^2=3x^6+7x^5+9x^4+6x^3+x^2+11$
- $y^2=12x^5+12x^4+7x^3+6x^2+2x+5$
- $y^2=11x^6+2x^5+x^4+11x^3+9x+5$
- $y^2=5x^6+8x^5+x^3+3x+2$
- $y^2=7x^6+6x^5+10x^3+4x^2+2x+12$
- $y^2=4x^6+3x^4+3x^3+12x^2+3x+8$
- $y^2=2x^6+x^5+8x^4+x^3+11x^2+10x+12$
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 4.0.1088.2. |
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.i_bi | $2$ | 2.169.e_ago |