Invariants
| Base field: | $\F_{61}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 5 x + 127 x^{2} + 305 x^{3} + 3721 x^{4}$ |
| Frobenius angles: | $\pm0.528198159238$, $\pm0.574402884533$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.1397525.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $27$ |
| Isomorphism classes: | 27 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $4159$ | $14718701$ | $51324642739$ | $191549542781525$ | $713409131034631504$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $67$ | $3951$ | $226117$ | $13834443$ | $844674702$ | $51520841751$ | $3142737237637$ | $191707300134483$ | $11694146433862987$ | $713342911559666806$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 27 curves (of which all are hyperelliptic):
- $y^2=45 x^6+43 x^5+56 x^4+x^3+22 x^2+23 x+30$
- $y^2=29 x^6+32 x^5+43 x^4+3 x^3+18 x^2+53 x+21$
- $y^2=49 x^6+58 x^5+15 x^4+21 x^3+x^2+19 x+49$
- $y^2=56 x^6+33 x^5+38 x^4+8 x^3+55 x^2+58 x+20$
- $y^2=5 x^6+40 x^5+25 x^4+6 x^3+35 x^2+47 x+31$
- $y^2=37 x^6+20 x^5+21 x^4+14 x^3+42 x^2+25 x+56$
- $y^2=17 x^6+32 x^5+23 x^4+31 x^3+2 x^2+17 x+20$
- $y^2=7 x^6+59 x^5+48 x^4+58 x^3+55 x^2+11 x+5$
- $y^2=39 x^6+18 x^5+24 x^4+34 x^3+30 x^2+10 x+31$
- $y^2=26 x^6+58 x^5+51 x^4+7 x^3+56 x^2+26 x+39$
- $y^2=25 x^6+28 x^5+34 x^4+15 x^3+54 x^2+47 x+58$
- $y^2=16 x^6+57 x^5+5 x^4+33 x^3+29 x^2+41 x+59$
- $y^2=23 x^6+20 x^5+20 x^4+37 x^3+45 x^2+52 x+56$
- $y^2=54 x^6+54 x^5+33 x^4+19 x^3+2 x^2+35 x+39$
- $y^2=32 x^6+49 x^5+9 x^4+31 x^3+33 x^2+30 x+48$
- $y^2=31 x^6+19 x^5+27 x^4+21 x^3+9 x^2+57 x+27$
- $y^2=x^6+43 x^5+6 x^4+41 x^3+7 x^2+56 x+9$
- $y^2=41 x^6+51 x^5+3 x^4+50 x^3+32 x^2+22 x+2$
- $y^2=43 x^6+14 x^5+48 x^4+47 x^3+31 x^2+41 x+38$
- $y^2=13 x^6+8 x^5+8 x^4+19 x^3+45 x^2+46 x+54$
- $y^2=42 x^6+6 x^5+17 x^4+50 x^3+28 x^2+40 x+23$
- $y^2=13 x^6+12 x^5+28 x^4+55 x^3+55 x^2+10 x+40$
- $y^2=14 x^6+19 x^5+59 x^4+60 x^3+51 x^2+10 x+21$
- $y^2=46 x^6+47 x^5+58 x^3+47 x^2+36 x+29$
- $y^2=53 x^6+27 x^5+33 x^4+29 x^3+8 x^2+23 x+33$
- $y^2=41 x^6+45 x^5+14 x^4+29 x^3+20 x^2+24 x+51$
- $y^2=58 x^6+49 x^5+10 x^4+37 x^3+18 x^2+28 x+55$
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 4.0.1397525.1. |
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.af_ex | $2$ | (not in LMFDB) |