Invariants
Base field: | $\F_{5}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 3 x + 5 x^{2} - 15 x^{3} + 25 x^{4}$ |
Frobenius angles: | $\pm0.113143297209$, $\pm0.585923223955$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.37845.1 |
Galois group: | $D_{4}$ |
Jacobians: | $2$ |
Isomorphism classes: | 2 |
This isogeny class is simple and geometrically 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $13$ | $637$ | $12649$ | $372645$ | $10187008$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $3$ | $27$ | $99$ | $595$ | $3258$ | $15747$ | $78039$ | $392515$ | $1957743$ | $9765702$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=2x^6+x^5+4x^4+4x^3+2x^2+x+3$
- $y^2=4x^6+x^5+x^4+x^3+4x^2+4x+2$
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 4.0.37845.1. |
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 |
---|---|---|
2.5.d_f | $2$ | 2.25.b_ap |