Invariants
| Base field: | $\F_{61}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 24 x + 255 x^{2} - 1464 x^{3} + 3721 x^{4}$ |
| Frobenius angles: | $\pm0.0628885257501$, $\pm0.312375219403$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-89 +24 \sqrt{11}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $8$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2489$ | $13602385$ | $51552775316$ | $191707783848265$ | $713306406880761329$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $38$ | $3656$ | $227126$ | $13845876$ | $844553078$ | $51519784718$ | $3142739220638$ | $191707313180836$ | $11694146316760046$ | $713342913891441176$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=2 x^6+53 x^5+42 x^4+22 x^3+53 x^2+52 x+51$
- $y^2=26 x^6+13 x^5+58 x^4+44 x^3+56 x^2+10 x+38$
- $y^2=54 x^6+38 x^5+35 x^4+37 x^3+46 x^2+38 x+45$
- $y^2=46 x^6+4 x^4+34 x^3+31 x^2+13 x+44$
- $y^2=17 x^6+x^5+38 x^4+4 x^3+6 x^2+6 x+44$
- $y^2=52 x^6+50 x^5+12 x^4+22 x^3+18 x^2+43 x+10$
- $y^2=51 x^6+24 x^5+42 x^4+5 x^3+60 x^2+41 x+51$
- $y^2=14 x^6+25 x^5+14 x^4+55 x^3+30 x^2+42 x+13$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{61}$.
Endomorphism algebra over $\F_{61}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-89 +24 \sqrt{11}})\). |
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 |
|---|---|---|
| 2.61.y_jv | $2$ | (not in LMFDB) |