Invariants
| Base field: | $\F_{5}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 4 x + 12 x^{2} - 20 x^{3} + 25 x^{4}$ |
| Frobenius angles: | $\pm0.223508181938$, $\pm0.458185759261$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-14 +4 \sqrt{2}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $2$ |
| Isomorphism classes: | 2 |
| Cyclic group of points: | yes |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $14$ | $868$ | $18494$ | $392336$ | $9877294$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $2$ | $34$ | $146$ | $630$ | $3162$ | $15922$ | $78458$ | $389022$ | $1947746$ | $9762594$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which all are hyperelliptic):
- $y^2=2 x^6+2 x^5+3 x^4+3 x^3+4 x^2+x$
- $y^2=2 x^6+4 x^4+x^3+3 x^2+2 x+1$
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 \(\Q(\sqrt{-14 +4 \sqrt{2}})\). |
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.e_m | $2$ | 2.25.i_bi |