Invariants
| Base field: | $\F_{67}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 26 x + 301 x^{2} - 1742 x^{3} + 4489 x^{4}$ |
| Frobenius angles: | $\pm0.157214918842$, $\pm0.249726292109$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-97 -26 \sqrt{2}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $10$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $3023$ | $19827857$ | $90662224676$ | $406319920746089$ | $1822985910876725703$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $42$ | $4416$ | $301440$ | $20163636$ | $1350234802$ | $90458977278$ | $6060712825062$ | $406067665053924$ | $27206534248344912$ | $1822837803921722416$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):
- $y^2=53 x^6+10 x^5+11 x^4+48 x^3+6 x^2+47 x+39$
- $y^2=63 x^6+17 x^5+35 x^4+10 x^3+30 x^2+27 x+23$
- $y^2=23 x^6+44 x^5+9 x^4+3 x^3+27 x^2+31 x+32$
- $y^2=7 x^6+65 x^5+24 x^4+6 x^3+22 x^2+36 x+49$
- $y^2=50 x^6+48 x^5+31 x^4+6 x^3+42 x^2+39 x+17$
- $y^2=2 x^6+9 x^5+48 x^4+5 x^3+14 x^2+27 x+5$
- $y^2=36 x^6+34 x^5+54 x^4+34 x^3+59 x^2+58 x+56$
- $y^2=36 x^6+57 x^5+22 x^4+44 x^3+45 x^2+35 x+19$
- $y^2=20 x^6+46 x^5+65 x^4+32 x^3+5 x^2+15 x+20$
- $y^2=16 x^6+62 x^5+54 x^4+49 x^3+57 x^2+10 x+30$
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 \(\Q(\sqrt{-97 -26 \sqrt{2}})\). |
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 |
|---|---|---|
| 2.67.ba_lp | $2$ | (not in LMFDB) |