Invariants
| Base field: | $\F_{67}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 28 x + 325 x^{2} - 1876 x^{3} + 4489 x^{4}$ |
| Frobenius angles: | $\pm0.0408519366732$, $\pm0.244784846033$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-98 +10 \sqrt{5}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $6$ |
| Isomorphism classes: | 6 |
| Cyclic group of points: | yes |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2911$ | $19559009$ | $90374021056$ | $406090579771961$ | $1822833985799040511$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $40$ | $4356$ | $300484$ | $20152260$ | $1350122280$ | $90458009382$ | $6060705478168$ | $406067616738564$ | $27206533998059068$ | $1822837803345932836$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=65 x^6+48 x^5+49 x^4+36 x^3+30 x^2+6 x+15$
- $y^2=35 x^6+8 x^5+24 x^4+20 x^3+37 x^2+46 x+61$
- $y^2=38 x^6+3 x^4+45 x^3+x^2+21 x+43$
- $y^2=51 x^6+24 x^5+59 x^4+x^3+58 x^2+50 x+15$
- $y^2=51 x^6+21 x^4+8 x^3+59 x^2+48 x+12$
- $y^2=7 x^6+18 x^5+37 x^4+14 x^3+47 x^2+51 x+30$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{67}$.
Endomorphism algebra over $\F_{67}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-98 +10 \sqrt{5}})\). |
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 |
|---|---|---|
| 2.67.bc_mn | $2$ | (not in LMFDB) |