Invariants
| Base field: | $\F_{61}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 10 x + 145 x^{2} + 610 x^{3} + 3721 x^{4}$ |
| Frobenius angles: | $\pm0.573727554125$, $\pm0.634690486021$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.3000896.2 |
| Galois group: | $D_{4}$ |
| Jacobians: | $21$ |
| Isomorphism classes: | 21 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $4487$ | $14569289$ | $51176370812$ | $191657962375481$ | $713429867187911727$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $72$ | $3912$ | $225462$ | $13842276$ | $844699252$ | $51520082142$ | $3142738661652$ | $191707347582468$ | $11694146147443182$ | $713342909735656552$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 21 curves (of which all are hyperelliptic):
- $y^2=x^6+24 x^5+35 x^4+41 x^3+21 x^2+28 x+16$
- $y^2=47 x^6+20 x^5+34 x^4+18 x^3+30 x^2+19 x+41$
- $y^2=43 x^6+26 x^5+42 x^4+53 x^3+6 x^2+16 x+23$
- $y^2=36 x^6+42 x^5+34 x^4+47 x^3+42 x^2+48 x+28$
- $y^2=7 x^6+39 x^5+39 x^4+39 x^3+15 x^2+2 x+58$
- $y^2=43 x^6+32 x^5+34 x^4+36 x^3+27 x^2+44 x+39$
- $y^2=38 x^6+31 x^5+45 x^4+27 x^3+37 x^2+9 x+54$
- $y^2=2 x^6+41 x^5+31 x^4+17 x^3+23 x^2+31 x+4$
- $y^2=42 x^6+60 x^5+30 x^4+5 x^3+38 x^2+56 x+42$
- $y^2=50 x^6+41 x^5+59 x^4+4 x^3+25 x^2+47 x+5$
- $y^2=27 x^6+25 x^5+19 x^4+19 x^3+5 x^2+x+59$
- $y^2=15 x^6+30 x^5+18 x^4+13 x^3+50 x^2+42 x+57$
- $y^2=28 x^6+47 x^5+7 x^3+51 x^2+56 x+29$
- $y^2=17 x^6+11 x^5+23 x^4+44 x^3+49 x^2+54 x+49$
- $y^2=28 x^6+2 x^5+51 x^4+58 x^3+36 x^2+35 x+47$
- $y^2=9 x^6+34 x^5+12 x^4+25 x^3+32 x^2+57 x+17$
- $y^2=6 x^6+47 x^5+57 x^4+25 x^3+29 x^2+45 x+41$
- $y^2=3 x^6+x^5+35 x^4+47 x^3+12 x^2+38 x+1$
- $y^2=17 x^6+46 x^5+25 x^4+28 x^3+45 x^2+4 x+5$
- $y^2=35 x^6+59 x^5+32 x^4+18 x^3+24 x^2+59 x+7$
- $y^2=22 x^6+3 x^5+14 x^4+58 x^3+59 x^2+27 x+36$
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 4.0.3000896.2. |
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.ak_fp | $2$ | (not in LMFDB) |