Invariants
| Base field: | $\F_{7^{2}}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 23 x + 229 x^{2} - 1127 x^{3} + 2401 x^{4}$ |
| Frobenius angles: | $\pm0.142622734692$, $\pm0.234081986531$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.81125.1 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1481$ | $5599661$ | $13871225201$ | $33266158087445$ | $79806326823818496$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $27$ | $2331$ | $117903$ | $5770563$ | $282525022$ | $13841568291$ | $678224025303$ | $33232930367523$ | $1628413572435627$ | $79792266156487006$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(6a+1)x^6+(a+4)x^5+(3a+2)x^4+4x^3+(5a+2)x^2+ax+3a+3$
- $y^2=(a+1)x^6+(6a+6)x^5+3ax^4+(2a+6)x^3+(2a+5)x^2+6ax+6a+1$
- $y^2=5x^6+(3a+4)x^5+(a+1)x^4+3ax^3+(3a+5)x^2+6x+5a+1$
- $y^2=(5a+6)x^6+(5a+6)x^5+(5a+5)x^4+(a+6)x^3+6ax^2+6x+4a+3$
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 4.0.81125.1. |
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_iv | $2$ | (not in LMFDB) |