Invariants
| Base field: | $\F_{61}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 26 x + 288 x^{2} - 1586 x^{3} + 3721 x^{4}$ |
| Frobenius angles: | $\pm0.107873325280$, $\pm0.243518934227$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.454464.1 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2398$ | $13481556$ | $51550019422$ | $191788022467536$ | $713387030318426398$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $36$ | $3622$ | $227112$ | $13851670$ | $844648536$ | $51520632262$ | $3142743257508$ | $191707310837854$ | $11694146129967732$ | $713342912959102102$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=2x^6+39x^5+50x^4+57x^3+15x^2+26x+15$
- $y^2=8x^6+15x^5+45x^4+40x^3+31x^2+39x+2$
- $y^2=11x^6+42x^5+3x^4+40x^3+24x^2+16x+18$
- $y^2=32x^6+49x^5+17x^4+5x^3+55x^2+50x+8$
- $y^2=46x^6+23x^5+21x^4+20x^3+38x^2+14x+60$
- $y^2=35x^6+16x^5+36x^4+15x^3+52x^2+55x+43$
- $y^2=44x^6+60x^5+22x^4+54x^3+33x^2+34x+40$
- $y^2=17x^6+39x^5+16x^4+43x^3+49x^2+50x+7$
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 4.0.454464.1. |
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.ba_lc | $2$ | (not in LMFDB) |