Invariants
Base field: | $\F_{71}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 29 x + 349 x^{2} - 2059 x^{3} + 5041 x^{4}$ |
Frobenius angles: | $\pm0.0815109748440$, $\pm0.228392276737$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.378053.1 |
Galois group: | $D_{4}$ |
Jacobians: | $7$ |
Isomorphism classes: | 7 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $3303$ | $24703137$ | $128027777877$ | $645867197632413$ | $3255337942816632048$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $43$ | $4899$ | $357709$ | $25416155$ | $1804281668$ | $128100542895$ | $9095119643555$ | $645753511104163$ | $45848500583521225$ | $3255243551760860574$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 7 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=67x^6+15x^5+13x^4+53x^3+55x^2+69x+23$
- $y^2=26x^6+13x^5+31x^4+31x^3+42x^2+58x+6$
- $y^2=63x^6+26x^5+12x^4+23x^3+69x^2+56x+34$
- $y^2=17x^6+27x^5+58x^4+14x^3+21x^2+68x+47$
- $y^2=26x^6+45x^5+40x^4+60x^3+48x^2+35x+27$
- $y^2=23x^6+56x^5+32x^4+21x^3+18x+23$
- $y^2=53x^6+40x^5+27x^4+66x^3+31x^2+42x+59$
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 4.0.378053.1. |
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.bd_nl | $2$ | (not in LMFDB) |