Invariants
Base field: | $\F_{71}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 27 x + 316 x^{2} - 1917 x^{3} + 5041 x^{4}$ |
Frobenius angles: | $\pm0.0761553981579$, $\pm0.282792842051$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.2970792.1 |
Galois group: | $D_{4}$ |
Jacobians: | $12$ |
Isomorphism classes: | 12 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $3414$ | $24929028$ | $128158023096$ | $645840715904928$ | $3255228403799473794$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $45$ | $4945$ | $358074$ | $25415113$ | $1804220955$ | $128099772538$ | $9095114765061$ | $645753513830449$ | $45848501014534326$ | $3255243556746294265$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 12 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=55x^6+30x^5+59x^4+28x^3+9x^2+x+6$
- $y^2=61x^6+53x^5+40x^4+28x^3+65x^2+4x+68$
- $y^2=61x^6+44x^5+39x^4+19x^3+29x^2+42x+35$
- $y^2=27x^6+58x^5+57x^4+42x^3+66x^2+40x+42$
- $y^2=67x^6+25x^5+19x^4+34x^3+68x^2+11x+33$
- $y^2=24x^6+11x^5+24x^4+24x^3+16x^2+3x+53$
- $y^2=61x^6+33x^5+56x^4+30x^3+31x^2+15x+35$
- $y^2=52x^6+54x^5+4x^4+41x^3+29x^2+9x+66$
- $y^2=9x^6+41x^5+24x^4+30x^3+48x^2+25x+70$
- $y^2=23x^6+51x^5+2x^4+6x^3+25x^2+40x+61$
- $y^2=41x^6+17x^5+4x^4+x^3+22x^2+24x+41$
- $y^2=61x^6+60x^5+55x^4+29x^3+40x^2+57x+16$
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 4.0.2970792.1. |
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_me | $2$ | (not in LMFDB) |