Invariants
| Base field: | $\F_{71}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 27 x + 315 x^{2} - 1917 x^{3} + 5041 x^{4}$ |
| Frobenius angles: | $\pm0.0612390793144$, $\pm0.286887128510$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.50653.1 |
| Galois group: | $C_4$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $3413$ | $24918313$ | $128128904003$ | $645798673326877$ | $3255187240167623408$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $45$ | $4943$ | $357993$ | $25413459$ | $1804198140$ | $128099535779$ | $9095112788499$ | $645753499132435$ | $45848500896055059$ | $3255243555564245918$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 7 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=68x^6+62x^5+32x^4+26x^3+30x^2+28x+51$
- $y^2=20x^6+58x^5+32x^4+11x^3+59x^2+27x+17$
- $y^2=32x^6+17x^5+68x^4+66x^3+8x^2+31x+19$
- $y^2=63x^6+19x^5+29x^4+29x^3+36x^2+58x+42$
- $y^2=55x^6+66x^5+34x^4+66x^3+5x^2+70x+33$
- $y^2=44x^6+15x^5+38x^4+30x^3+10x^2+6x+64$
- $y^2=68x^6+47x^5+16x^4+24x^3+51x^2+52x+40$
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 4.0.50653.1. |
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.bb_md | $2$ | (not in LMFDB) |