Invariants
Base field: | $\F_{71}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 25 x + 280 x^{2} - 1775 x^{3} + 5041 x^{4}$ |
Frobenius angles: | $\pm0.0310856259029$, $\pm0.337638390561$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.3112136.1 |
Galois group: | $D_{4}$ |
Jacobians: | $8$ |
Isomorphism classes: | 8 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $3522$ | $25083684$ | $128117525832$ | $645632554526496$ | $3255022765603836102$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $47$ | $4977$ | $357962$ | $25406921$ | $1804106977$ | $128098974042$ | $9095112973567$ | $645753525176113$ | $45848500870314182$ | $3255243550356052377$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=67x^6+3x^5+2x^4+30x^3+37x^2+8x+50$
- $y^2=66x^6+42x^5+15x^4+50x^3+16x^2+57x+35$
- $y^2=67x^6+33x^5+69x^4+70x^3+13x^2+26x+10$
- $y^2=45x^6+65x^5+7x^4+65x^3+38x^2+21x+34$
- $y^2=26x^6+36x^5+58x^4+28x^3+51x^2+59x+66$
- $y^2=67x^6+26x^5+20x^4+67x^3+35x^2+29x+10$
- $y^2=4x^6+26x^5+2x^4+59x^3+49x^2+29x+41$
- $y^2=11x^6+15x^5+68x^4+18x^3+20x^2+29x+2$
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 4.0.3112136.1. |
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.z_ku | $2$ | (not in LMFDB) |