Invariants
| Base field: | $\F_{7^{2}}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 23 x + 226 x^{2} - 1127 x^{3} + 2401 x^{4}$ |
| Frobenius angles: | $\pm0.0798729527170$, $\pm0.264499436907$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.373388.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $4$ |
| Isomorphism classes: | 4 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1478$ | $5583884$ | $13846660736$ | $33245283888128$ | $79794339197567238$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $27$ | $2325$ | $117696$ | $5766945$ | $282482587$ | $13841208354$ | $678221905899$ | $33232924672833$ | $1628413619537856$ | $79792267001872405$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(4a+5)x^6+5x^5+(a+1)x^4+(5a+3)x^3+4x^2+ax+6a+1$
- $y^2=(5a+4)x^6+(6a+3)x^5+(2a+2)x^4+(4a+6)x^3+(2a+2)x^2+4ax+4a+2$
- $y^2=4x^6+(6a+2)x^5+(3a+6)x^4+(a+1)x^3+6ax^2+(5a+6)x+6a+6$
- $y^2=2ax^6+(5a+4)x^5+5ax^4+(6a+6)x^3+3x^2+(a+3)x+3a+4$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{7^{2}}$.
Endomorphism algebra over $\F_{7^{2}}$| The endomorphism algebra of this simple isogeny class is 4.0.373388.1. |
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 |
|---|---|---|
| 2.49.x_is | $2$ | (not in LMFDB) |