Invariants
Base field: | $\F_{5}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 2 x + x^{2} - 10 x^{3} + 25 x^{4}$ |
Frobenius angles: | $\pm0.119187091912$, $\pm0.660634678170$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.65600.5 |
Galois group: | $D_{4}$ |
Jacobians: | $2$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $15$ | $585$ | $12060$ | $403065$ | $10080375$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $4$ | $24$ | $94$ | $644$ | $3224$ | $15534$ | $78824$ | $392644$ | $1953094$ | $9772824$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=2x^6+3x^5+4x^4+3x+2$
- $y^2=4x^6+x^5+4x^3+4x^2+2x+2$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{5}$.
Endomorphism algebra over $\F_{5}$The endomorphism algebra of this simple isogeny class is 4.0.65600.5. |
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 |
---|---|---|
2.5.c_b | $2$ | 2.25.ac_l |