Invariants
| Base field: | $\F_{61}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 27 x + 301 x^{2} - 1647 x^{3} + 3721 x^{4}$ |
| Frobenius angles: | $\pm0.0643104250482$, $\pm0.230612073966$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.2197.1 |
| Galois group: | $C_4$ |
| Jacobians: | $9$ |
| Isomorphism classes: | 9 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2349$ | $13382253$ | $51465248721$ | $191736116755173$ | $713360334091993344$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $35$ | $3595$ | $226739$ | $13847923$ | $844616930$ | $51520377283$ | $3142741037471$ | $191707289788099$ | $11694145935604151$ | $713342911430554390$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 9 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=43x^6+18x^5+9x^4+51x^3+58x^2+28x+55$
- $y^2=40x^6+23x^5+41x^4+46x^3+47x^2+47x+11$
- $y^2=7x^6+53x^5+56x^4+46x^3+14x^2+9x+51$
- $y^2=35x^6+59x^5+39x^4+23x^3+27x^2+51x+2$
- $y^2=16x^6+33x^5+3x^4+33x^3+25x^2+33x+5$
- $y^2=46x^6+14x^5+51x^4+6x^3+9x^2+32x+19$
- $y^2=38x^6+50x^5+5x^4+18x^3+29x^2+56x+4$
- $y^2=19x^6+32x^5+21x^4+58x^3+6x^2+11x+12$
- $y^2=37x^6+47x^5+58x^4+59x^3+48x^2+31x+45$
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 4.0.2197.1. |
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.bb_lp | $2$ | (not in LMFDB) |