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: | 4.0.3068560.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $8$ |
| Isomorphism classes: | 8 |
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 contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=2x^6+53x^5+42x^4+22x^3+53x^2+52x+51$
- $y^2=26x^6+13x^5+58x^4+44x^3+56x^2+10x+38$
- $y^2=54x^6+38x^5+35x^4+37x^3+46x^2+38x+45$
- $y^2=46x^6+4x^4+34x^3+31x^2+13x+44$
- $y^2=17x^6+x^5+38x^4+4x^3+6x^2+6x+44$
- $y^2=52x^6+50x^5+12x^4+22x^3+18x^2+43x+10$
- $y^2=51x^6+24x^5+42x^4+5x^3+60x^2+41x+51$
- $y^2=14x^6+25x^5+14x^4+55x^3+30x^2+42x+13$
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 4.0.3068560.1. |
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 |
|---|---|---|
| 2.61.y_jv | $2$ | (not in LMFDB) |