Invariants
| Base field: | $\F_{11}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 7 x + 31 x^{2} - 77 x^{3} + 121 x^{4}$ |
| Frobenius angles: | $\pm0.205138408518$, $\pm0.417639346619$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.110357.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $5$ |
| Isomorphism classes: | 5 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $69$ | $16353$ | $1878111$ | $215548893$ | $25941638544$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $5$ | $135$ | $1409$ | $14723$ | $161080$ | $1773603$ | $19497511$ | $214360339$ | $2357791151$ | $25936832790$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 5 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=7x^6+8x^5+6x^4+4x^3+2x^2+3x+6$
- $y^2=10x^6+x^5+7x^4+4x^3+10x^2+x+8$
- $y^2=4x^6+4x^5+4x^3+6x^2+x+5$
- $y^2=6x^6+10x^5+6x^3+6x^2+10x+2$
- $y^2=8x^6+7x^5+x^4+5x^2+8$
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 4.0.110357.1. |
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 |
|---|---|---|
| 2.11.h_bf | $2$ | 2.121.n_ev |