Invariants
Base field: | $\F_{71}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 30 x + 365 x^{2} - 2130 x^{3} + 5041 x^{4}$ |
Frobenius angles: | $\pm0.0727378648374$, $\pm0.201537683200$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.92736.1 |
Galois group: | $D_{4}$ |
Jacobians: | $4$ |
Isomorphism classes: | 4 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $3247$ | $24570049$ | $127907369572$ | $645807251462121$ | $3255333131609617327$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $42$ | $4872$ | $357372$ | $25413796$ | $1804279002$ | $128100708582$ | $9095122000902$ | $645753526617988$ | $45848500561339812$ | $3255243549769577352$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=12x^6+4x^5+32x^4+37x^3+56x^2+37x+9$
- $y^2=62x^6+42x^5+29x^4+3x^3+6x^2+28x+56$
- $y^2=38x^6+42x^5+33x^4+37x^3+19x^2+27x+53$
- $y^2=x^6+5x^5+64x^4+56x^3+28x^2+18x+52$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{71}$.
Endomorphism algebra over $\F_{71}$The endomorphism algebra of this simple isogeny class is 4.0.92736.1. |
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 |
---|---|---|
2.71.be_ob | $2$ | (not in LMFDB) |