Invariants
Base field: | $\F_{71}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 26 x + 298 x^{2} - 1846 x^{3} + 5041 x^{4}$ |
Frobenius angles: | $\pm0.0545373375450$, $\pm0.311776771058$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.273104.1 |
Galois group: | $D_{4}$ |
Jacobians: | $30$ |
Isomorphism classes: | 40 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $3468$ | $25011216$ | $128146307292$ | $645737973970176$ | $3255114975278762748$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $46$ | $4962$ | $358042$ | $25411070$ | $1804158086$ | $128099256642$ | $9095112901298$ | $645753520218814$ | $45848501057255422$ | $3255243554886800802$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 30 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=53x^6+60x^5+21x^4+23x^3+21x^2+15x+54$
- $y^2=63x^6+29x^5+38x^4+70x^3+38x^2+44$
- $y^2=6x^6+37x^5+22x^4+42x^3+62x^2+49x+9$
- $y^2=14x^6+5x^5+46x^4+41x^3+40x^2+62x+61$
- $y^2=3x^6+31x^5+2x^4+x^3+46x^2+68x+65$
- $y^2=14x^6+18x^5+68x^4+67x^3+64x^2+25x+49$
- $y^2=46x^6+17x^5+47x^4+5x^3+24x^2+59x+24$
- $y^2=28x^6+18x^5+16x^4+25x^3+45x^2+51x+53$
- $y^2=63x^6+20x^4+47x^3+2x^2+32x+56$
- $y^2=63x^6+55x^5+43x^4+11x^3+23x^2+16x+28$
- $y^2=37x^6+16x^5+42x^4+4x^3+49x^2+12x+55$
- $y^2=9x^6+20x^5+25x^4+25x^3+23x^2+32x+44$
- $y^2=14x^6+62x^4+39x^3+49x^2+52x+18$
- $y^2=8x^6+47x^5+53x^4+33x^3+14x^2+x+20$
- $y^2=23x^6+67x^5+18x^4+8x^3+28x^2+57x+63$
- $y^2=68x^6+21x^5+59x^4+52x^3+50x^2+28x+37$
- $y^2=63x^6+17x^5+2x^4+48x^3+60x^2+39x+26$
- $y^2=20x^6+8x^5+21x^4+2x^3+43x^2+40x+31$
- $y^2=49x^6+36x^5+50x^4+38x^3+67x^2+45x+38$
- $y^2=21x^6+53x^5+22x^4+43x^3+37x^2+49x+23$
- $y^2=39x^6+52x^5+7x^4+12x^3+x^2+17x+47$
- $y^2=51x^6+64x^5+65x^4+10x^3+47x^2+59x+54$
- $y^2=34x^6+44x^5+64x^4+55x^3+18x^2+51x+23$
- $y^2=69x^6+10x^5+15x^4+30x^3+19x^2+12x+39$
- $y^2=49x^6+20x^5+30x^4+66x^3+41x^2+20x+52$
- $y^2=61x^6+47x^5+43x^4+45x^3+11x^2+55x+54$
- $y^2=68x^6+53x^5+5x^4+48x^3+16x^2+50x+61$
- $y^2=13x^6+15x^5+7x^4+9x^3+16x^2+3x+61$
- $y^2=52x^6+14x^5+23x^4+45x^3+24x^2+61x+41$
- $y^2=44x^6+20x^5+9x^4+33x^3+6x^2+17x+16$
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 4.0.273104.1. |
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_lm | $2$ | (not in LMFDB) |