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 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2495$ | $13650145$ | $51651330320$ | $191813489207545$ | $713379988161809375$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $38$ | $3668$ | $227558$ | $13853508$ | $844640198$ | $51520471238$ | $3142742783918$ | $191707322200708$ | $11694146305804958$ | $713342913935332148$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 28 curves (of which all are hyperelliptic):
- $y^2=17 x^6+4 x^5+34 x^4+4 x^3+12 x^2+29 x+53$
- $y^2=60 x^6+30 x^5+45 x^4+16 x^3+31 x^2+25 x+42$
- $y^2=44 x^6+34 x^5+24 x^4+51 x^3+40 x^2+31 x+10$
- $y^2=22 x^6+24 x^5+45 x^4+11 x^3+9 x^2+14 x+18$
- $y^2=35 x^6+56 x^5+44 x^4+49 x^3+16 x^2+42 x+45$
- $y^2=30 x^6+24 x^5+29 x^4+26 x^3+24 x^2+60 x+57$
- $y^2=21 x^6+31 x^5+47 x^4+14 x^3+48 x^2+6 x+40$
- $y^2=37 x^6+22 x^5+60 x^4+32 x^3+8 x^2+44 x+10$
- $y^2=20 x^6+54 x^5+42 x^4+49 x^3+44 x^2+49 x+3$
- $y^2=8 x^6+57 x^5+2 x^4+36 x^3+53 x^2+50 x+45$
- $y^2=3 x^6+22 x^5+26 x^4+3 x^3+9 x^2+28 x+29$
- $y^2=26 x^6+44 x^5+30 x^4+6 x^3+2 x^2+53 x+51$
- $y^2=26 x^6+29 x^5+30 x^4+46 x^3+49 x^2+38 x+52$
- $y^2=37 x^6+20 x^5+8 x^4+60 x^3+34 x^2+32$
- $y^2=30 x^6+45 x^5+3 x^4+11 x^3+23 x^2+49 x+24$
- $y^2=51 x^6+55 x^5+31 x^4+28 x^3+29 x^2+30 x+7$
- $y^2=44 x^6+60 x^5+18 x^4+60 x^3+57 x^2+31 x+3$
- $y^2=44 x^6+30 x^5+50 x^4+18 x^3+46 x^2+29 x+38$
- $y^2=6 x^6+46 x^5+11 x^4+21 x^3+58 x^2+41 x+41$
- $y^2=6 x^6+35 x^5+17 x^4+10 x^3+32 x^2+50 x+57$
- $y^2=56 x^6+4 x^5+51 x^4+26 x^3+42 x^2+46 x+51$
- $y^2=29 x^6+23 x^5+36 x^4+15 x^3+25 x^2+35 x+27$
- $y^2=32 x^6+27 x^5+21 x^4+5 x^3+36 x^2+39 x+21$
- $y^2=54 x^6+x^5+36 x^4+38 x^3+4 x^2+47 x+4$
- $y^2=15 x^6+30 x^5+49 x^4+32 x^3+25 x^2+8 x+29$
- $y^2=32 x^6+50 x^5+41 x^4+12 x^3+9 x^2+53 x+59$
- $y^2=52 x^6+44 x^5+35 x^4+29 x^3+57 x^2+2 x+27$
- $y^2=58 x^6+60 x^5+43 x^4+46 x^3+29 x^2+32 x+2$
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) |