Invariants
Base field: | $\F_{71}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 26 x + 303 x^{2} - 1846 x^{3} + 5041 x^{4}$ |
Frobenius angles: | $\pm0.111527558034$, $\pm0.293743666571$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.386624.1 |
Galois group: | $D_{4}$ |
Jacobians: | $18$ |
Isomorphism classes: | 18 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $3473$ | $25064641$ | $128286451952$ | $645928857247961$ | $3255286195902363673$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $46$ | $4972$ | $358432$ | $25418580$ | $1804252986$ | $128100121102$ | $9095118876358$ | $645753556102884$ | $45848501333485792$ | $3255243557923269852$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 18 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=70x^6+29x^5+24x^4+60x^3+59x^2+10x+46$
- $y^2=10x^6+37x^5+14x^4+32x^3+69x^2+34x+24$
- $y^2=45x^6+36x^5+2x^4+6x^3+6x^2+35x+39$
- $y^2=16x^6+39x^5+24x^4+58x^3+37x^2+48x+14$
- $y^2=63x^6+68x^5+13x^4+67x^3+67x^2+25x+23$
- $y^2=68x^6+58x^5+46x^4+3x^3+11x^2+41x+56$
- $y^2=27x^6+54x^5+63x^4+40x^3+28x^2+31x+56$
- $y^2=39x^6+54x^5+43x^4+12x^3+60x^2+12x+26$
- $y^2=2x^6+69x^5+39x^4+8x^3+35x^2+5x+13$
- $y^2=5x^6+48x^5+63x^4+59x^3+21x^2+50x+56$
- $y^2=31x^6+24x^5+61x^4+48x^3+45x^2+47x+67$
- $y^2=68x^6+63x^5+60x^4+62x^3+4x^2+9x+59$
- $y^2=44x^6+41x^5+9x^4+41x^3+9x^2+15x+9$
- $y^2=56x^6+70x^5+37x^4+27x^3+33x^2+68x+46$
- $y^2=47x^6+47x^5+59x^4+28x^3+53x^2+60x+7$
- $y^2=39x^6+68x^5+18x^4+16x^3+9x^2+48x+42$
- $y^2=21x^6+24x^5+53x^4+18x^3+68x^2+34x+42$
- $y^2=42x^6+36x^5+35x^4+2x^3+8x^2+31x+51$
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 4.0.386624.1. |
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.ba_lr | $2$ | (not in LMFDB) |