Invariants
| Base field: | $\F_{11}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 4 x + 6 x^{2} - 44 x^{3} + 121 x^{4}$ |
| Frobenius angles: | $\pm0.0702993117305$, $\pm0.621564005517$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\zeta_{5})\) |
| Galois group: | $C_4$ |
| Jacobians: | $10$ |
| Isomorphism classes: | 18 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $80$ | $14080$ | $1612880$ | $211988480$ | $26001250000$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $8$ | $118$ | $1208$ | $14478$ | $161448$ | $1769158$ | $19482968$ | $214393758$ | $2357958728$ | $25937365398$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):
- $y^2=2 x^5+9$
- $y^2=7 x^6+3 x^5+3 x^4+x^3+2 x^2+6 x+10$
- $y^2=9 x^6+9 x^5+10 x^4+7 x^3+9 x^2+9 x+8$
- $y^2=7 x^6+9 x^5+9 x^4+x^3+3 x^2+10 x+10$
- $y^2=4 x^6+7 x^5+5 x^4+7 x^3+7 x^2+10 x+6$
- $y^2=7 x^6+2 x^5+6 x^4+8 x^3+9 x^2+5 x+4$
- $y^2=2 x^6+2 x^5+5 x^4+9 x^2+9 x+6$
- $y^2=3 x^6+2 x^5+3 x^4+3 x^3+6 x^2+x+6$
- $y^2=7 x^6+4 x^5+6 x^4+3 x^3+5 x^2+5 x+5$
- $y^2=2 x^5+9 x^4+9 x^3+5 x^2+10$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{11}$.
Endomorphism algebra over $\F_{11}$| The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{5})\). |
Base change
This is a primitive isogeny class.