Invariants
Base field: | $\F_{5}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 2 x + 5 x^{2} - 10 x^{3} + 25 x^{4}$ |
Frobenius angles: | $\pm0.219592142625$, $\pm0.605066734301$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.83520.4 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $19$ | $817$ | $14668$ | $412585$ | $10465219$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $4$ | $32$ | $118$ | $660$ | $3344$ | $15662$ | $77648$ | $390820$ | $1950814$ | $9754352$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=x^6+3x^5+x^4+3x^3+2x^2+x+2$
- $y^2=3x^6+3x^5+x^4+4x^3+2x+3$
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 4.0.83520.4. |
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_f | $2$ | 2.25.g_bj |