Invariants
Base field: | $\F_{71}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 28 x + 330 x^{2} - 1988 x^{3} + 5041 x^{4}$ |
Frobenius angles: | $\pm0.0169447458234$, $\pm0.269320814396$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.2048.2 |
Galois group: | $C_4$ |
Jacobians: | $7$ |
Isomorphism classes: | 9 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $3356$ | $24794128$ | $128029685084$ | $645751757637632$ | $3255166712871654556$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $44$ | $4918$ | $357716$ | $25411614$ | $1804186764$ | $128099349334$ | $9095108896820$ | $645753440013630$ | $45848500236993260$ | $3255243549958976118$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 7 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=13x^6+19x^5+27x^4+13x^3+5x^2+49x$
- $y^2=40x^6+69x^5+13x^4+24x^3+32x^2+11x+58$
- $y^2=12x^6+22x^5+18x^4+24x^3+67x^2+21x+39$
- $y^2=3x^6+8x^5+6x^4+25x^3+39x^2+39x+47$
- $y^2=63x^6+15x^5+62x^4+46x^3+2x^2+5x+66$
- $y^2=54x^6+17x^5+66x^4+57x^3+48x^2+44x+28$
- $y^2=17x^6+42x^5+20x^4+67x^3+4x^2+28x+61$
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 4.0.2048.2. |
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.bc_ms | $2$ | (not in LMFDB) |