Invariants
| Base field: | $\F_{71}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 15 x + 71 x^{2} )^{2}$ |
| $1 - 30 x + 367 x^{2} - 2130 x^{3} + 5041 x^{4}$ | |
| Frobenius angles: | $\pm0.150643965450$, $\pm0.150643965450$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $12$ |
This isogeny class is not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $3249$ | $24591681$ | $127972183824$ | $645915871265625$ | $3255462501531660729$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $42$ | $4876$ | $357552$ | $25418068$ | $1804350702$ | $128101650766$ | $9095132045202$ | $645753612501988$ | $45848501093328912$ | $3255243550863887356$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 12 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=13x^6+66x^5+22x^4+55x^3+22x^2+66x+13$
- $y^2=70x^6+4x^5+62x^4+31x^3+44x^2+36x+44$
- $y^2=27x^6+56x^5+30x^4+7x^3+30x^2+56x+27$
- $y^2=x^6+51x^5+68x^4+17x^3+68x^2+51x+1$
- $y^2=57x^6+57x^5+27x^4+40x^3+27x^2+57x+57$
- $y^2=14x^6+22x^5+32x^4+27x^3+24x^2+39x+17$
- $y^2=25x^6+54x^5+33x^4+55x^3+33x^2+54x+25$
- $y^2=22x^6+70x^5+57x^4+64x^3+57x^2+70x+22$
- $y^2=13x^6+60x^5+26x^4+69x^3+26x^2+60x+13$
- $y^2=22x^6+8x^5+6x^4+16x^3+15x^2+50x+42$
- $y^2=65x^6+42x^4+28x^3+42x^2+65$
- $y^2=63x^6+41x^5+68x^4+21x^3+68x^2+41x+63$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{71}$.
Endomorphism algebra over $\F_{71}$| The isogeny class factors as 1.71.ap 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-59}) \)$)$ |
Base change
This is a primitive isogeny class.