Invariants
Base field: | $\F_{5^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 15 x + 102 x^{2} - 375 x^{3} + 625 x^{4}$ |
Frobenius angles: | $\pm0.0946070441600$, $\pm0.316968049676$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.174556.1 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $338$ | $377884$ | $245539424$ | $152731643584$ | $95346211805858$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $11$ | $605$ | $15716$ | $390993$ | $9763451$ | $244117442$ | $6103471091$ | $152588542369$ | $3814704249716$ | $95367467713565$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(3a+1)x^6+3x^5+(3a+2)x^4+3ax^3+(4a+2)x^2+(3a+4)x+3a+1$
- $y^2=(3a+2)x^6+(a+2)x^5+(a+4)x^4+(4a+4)x^3+(2a+1)x^2+(2a+2)x+4a+3$
- $y^2=(4a+1)x^6+(3a+1)x^5+(3a+4)x^4+(a+1)x^3+(3a+1)x^2+(3a+3)x+2a+4$
- $y^2=(3a+1)x^6+(3a+3)x^5+(2a+4)x^4+(a+2)x^3+(3a+3)x^2+(3a+3)x+3a+1$
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 4.0.174556.1. |
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.p_dy | $2$ | 2.625.av_po |