Invariants
Base field: | $\F_{5^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 13 x + 83 x^{2} - 325 x^{3} + 625 x^{4}$ |
Frobenius angles: | $\pm0.0967753949370$, $\pm0.387586574437$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.1080141.1 |
Galois group: | $D_{4}$ |
Jacobians: | $6$ |
Isomorphism classes: | 6 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $371$ | $388437$ | $245154203$ | $152342272341$ | $95304484355696$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $13$ | $623$ | $15691$ | $389995$ | $9759178$ | $244132175$ | $6103695247$ | $152589225907$ | $3814700999161$ | $95367433006958$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(a+2)x^6+ax^5+(3a+4)x^4+x^3+(2a+1)x^2+(3a+2)x+a+2$
- $y^2=(4a+3)x^6+x^5+2x^4+(3a+1)x^3+3x^2+(4a+2)x+4a+3$
- $y^2=(4a+3)x^6+(a+1)x^5+4x^4+2ax^3+x+a+4$
- $y^2=3x^6+(a+2)x^5+(3a+1)x^4+(3a+1)x^3+4x^2+(2a+1)x+4a$
- $y^2=(3a+4)x^6+(2a+1)x^5+(3a+3)x^3+(4a+4)x^2+2ax+3a+2$
- $y^2=(a+4)x^6+(3a+3)x^5+4x^4+(4a+1)x^3+ax^2+(4a+4)x+2a+3$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{5^{2}}$.
Endomorphism algebra over $\F_{5^{2}}$The endomorphism algebra of this simple isogeny class is 4.0.1080141.1. |
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 |
---|---|---|
2.25.n_df | $2$ | 2.625.ad_alz |