Invariants
| Base field: | $\F_{5}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - x - x^{2} - 5 x^{3} + 25 x^{4}$ |
| Frobenius angles: | $\pm0.169335136474$, $\pm0.720319309201$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-9 +2 \sqrt{5}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $3$ |
| Isomorphism classes: | 3 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $19$ | $589$ | $13471$ | $438805$ | $9968464$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $5$ | $23$ | $107$ | $699$ | $3190$ | $15743$ | $79147$ | $390259$ | $1952765$ | $9767078$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 3 curves (of which all are hyperelliptic):
- $y^2=4 x^6+3 x^5+x^4+4 x^3+3 x^2+4 x+3$
- $y^2=4 x^6+4 x^4+4 x^3+3 x^2+2 x+2$
- $y^2=2 x^6+2 x^5+4 x^4+3 x^2+3$
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 \(\Q(\sqrt{-9 +2 \sqrt{5}})\). |
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.b_ab | $2$ | 2.25.ad_bp |