Invariants
| Base field: | $\F_{7^{2}}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 23 x + 225 x^{2} - 1127 x^{3} + 2401 x^{4}$ |
| Frobenius angles: | $\pm0.0550320885809$, $\pm0.271501929914$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.284445.2 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1477$ | $5578629$ | $13838475301$ | $33238280483205$ | $79790213515005952$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $27$ | $2323$ | $117627$ | $5765731$ | $282467982$ | $13841074771$ | $678220924443$ | $33232918531651$ | $1628413582469883$ | $79792266739810078$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(3a+3)x^6+(3a+4)x^5+3x^4+(6a+4)x^3+(2a+1)x^2+(3a+3)x+2a+3$
- $y^2=(a+6)x^6+(a+5)x^5+(2a+2)x^4+(2a+3)x^3+2ax^2+(a+4)x+3a+1$
- $y^2=(2a+5)x^6+(2a+5)x^5+(4a+5)x^4+(6a+4)x^3+6x^2+(4a+5)x+2a+6$
- $y^2=(2a+6)x^6+4ax^5+(5a+1)x^4+3x^3+(3a+5)x^2+(4a+5)x+4a+3$
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 4.0.284445.2. |
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_ir | $2$ | (not in LMFDB) |