Invariants
Base field: | $\F_{5}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 3 x + 5 x^{2} )( 1 + 5 x^{2} )$ |
$1 - 3 x + 10 x^{2} - 15 x^{3} + 25 x^{4}$ | |
Frobenius angles: | $\pm0.265942140215$, $\pm0.5$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $2$ |
Isomorphism classes: | 10 |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
$p$-rank: | $1$ |
Slopes: | $[0, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $18$ | $972$ | $18144$ | $388800$ | $9950058$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $3$ | $37$ | $144$ | $625$ | $3183$ | $15802$ | $77619$ | $388225$ | $1952208$ | $9774877$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=2x^6+4x^5+2x^4+x^3+2x^2+2x$
- $y^2=4x^6+x^5+4x^4+x^3+3x^2+3x+2$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{5^{2}}$.
Endomorphism algebra over $\F_{5}$The isogeny class factors as 1.5.ad $\times$ 1.5.a and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
The base change of $A$ to $\F_{5^{2}}$ is 1.25.b $\times$ 1.25.k. The endomorphism algebra for each factor is:
|
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.d_k | $2$ | 2.25.l_ci |