Invariants
| Base field: | $\F_{61}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 24 x + 261 x^{2} - 1464 x^{3} + 3721 x^{4}$ |
| Frobenius angles: | $\pm0.135025069903$, $\pm0.285069722533$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.153625.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $28$ |
| Isomorphism classes: | 28 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2495$ | $13650145$ | $51651330320$ | $191813489207545$ | $713379988161809375$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $38$ | $3668$ | $227558$ | $13853508$ | $844640198$ | $51520471238$ | $3142742783918$ | $191707322200708$ | $11694146305804958$ | $713342913935332148$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 28 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=17x^6+4x^5+34x^4+4x^3+12x^2+29x+53$
- $y^2=60x^6+30x^5+45x^4+16x^3+31x^2+25x+42$
- $y^2=44x^6+34x^5+24x^4+51x^3+40x^2+31x+10$
- $y^2=22x^6+24x^5+45x^4+11x^3+9x^2+14x+18$
- $y^2=35x^6+56x^5+44x^4+49x^3+16x^2+42x+45$
- $y^2=30x^6+24x^5+29x^4+26x^3+24x^2+60x+57$
- $y^2=21x^6+31x^5+47x^4+14x^3+48x^2+6x+40$
- $y^2=37x^6+22x^5+60x^4+32x^3+8x^2+44x+10$
- $y^2=20x^6+54x^5+42x^4+49x^3+44x^2+49x+3$
- $y^2=8x^6+57x^5+2x^4+36x^3+53x^2+50x+45$
- $y^2=3x^6+22x^5+26x^4+3x^3+9x^2+28x+29$
- $y^2=26x^6+44x^5+30x^4+6x^3+2x^2+53x+51$
- $y^2=26x^6+29x^5+30x^4+46x^3+49x^2+38x+52$
- $y^2=37x^6+20x^5+8x^4+60x^3+34x^2+32$
- $y^2=30x^6+45x^5+3x^4+11x^3+23x^2+49x+24$
- $y^2=51x^6+55x^5+31x^4+28x^3+29x^2+30x+7$
- $y^2=44x^6+60x^5+18x^4+60x^3+57x^2+31x+3$
- $y^2=44x^6+30x^5+50x^4+18x^3+46x^2+29x+38$
- $y^2=6x^6+46x^5+11x^4+21x^3+58x^2+41x+41$
- $y^2=6x^6+35x^5+17x^4+10x^3+32x^2+50x+57$
- $y^2=56x^6+4x^5+51x^4+26x^3+42x^2+46x+51$
- $y^2=29x^6+23x^5+36x^4+15x^3+25x^2+35x+27$
- $y^2=32x^6+27x^5+21x^4+5x^3+36x^2+39x+21$
- $y^2=54x^6+x^5+36x^4+38x^3+4x^2+47x+4$
- $y^2=15x^6+30x^5+49x^4+32x^3+25x^2+8x+29$
- $y^2=32x^6+50x^5+41x^4+12x^3+9x^2+53x+59$
- $y^2=52x^6+44x^5+35x^4+29x^3+57x^2+2x+27$
- $y^2=58x^6+60x^5+43x^4+46x^3+29x^2+32x+2$
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 4.0.153625.1. |
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_kb | $2$ | (not in LMFDB) |