Invariants
Base field: | $\F_{71}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 26 x + 304 x^{2} - 1846 x^{3} + 5041 x^{4}$ |
Frobenius angles: | $\pm0.121180770286$, $\pm0.289393443965$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.5434688.1 |
Galois group: | $D_{4}$ |
Jacobians: | $32$ |
Isomorphism classes: | 32 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $3474$ | $25075332$ | $128314486314$ | $645966730966608$ | $3255319033139941434$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $46$ | $4974$ | $358510$ | $25420070$ | $1804271186$ | $128100268350$ | $9095119505714$ | $645753553951678$ | $45848501267585854$ | $3255243557258494974$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 32 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=33x^6+36x^5+49x^4+15x^3+7x^2+31x+66$
- $y^2=35x^6+45x^5+50x^4+7x^3+32x^2+61x+61$
- $y^2=52x^6+36x^5+37x^4+x^3+6x^2+53x+65$
- $y^2=x^6+62x^5+43x^4+31x^3+38x^2+19x+63$
- $y^2=31x^6+34x^5+63x^4+5x^3+50x^2+10x+36$
- $y^2=63x^6+2x^5+63x^4+15x^3+44x^2+13x+36$
- $y^2=55x^6+36x^5+19x^4+64x^3+56x^2+70x+42$
- $y^2=70x^6+6x^5+69x^4+19x^3+40x^2+61x+45$
- $y^2=21x^6+44x^5+25x^4+19x^3+8x^2+48x+41$
- $y^2=64x^6+31x^5+63x^4+67x^3+61x^2+11x+16$
- $y^2=8x^6+59x^5+23x^4+31x^3+62x^2+6x+16$
- $y^2=64x^6+45x^5+5x^4+55x^3+60x^2+32x+44$
- $y^2=70x^6+x^5+50x^4+64x^3+18x^2+6x+11$
- $y^2=23x^6+30x^5+70x^4+47x^3+51x^2+17x+13$
- $y^2=13x^6+38x^4+10x^3+60x^2+32x+2$
- $y^2=47x^6+56x^5+27x^4+48x^3+31x^2+46x+58$
- $y^2=9x^6+14x^5+45x^4+35x^3+23x^2+45x+26$
- $y^2=56x^6+50x^5+66x^4+13x^3+8x^2+52x+42$
- $y^2=46x^6+27x^5+19x^4+35x^3+x^2+18x+61$
- $y^2=30x^6+36x^5+7x^4+3x^3+57x^2+x+11$
- and 12 more
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 4.0.5434688.1. |
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_ls | $2$ | (not in LMFDB) |