Abelian variety isogeny class 2.71.i_gc over $\F_{71}$

• Label: 2.71.i_gc
• Base field: $\F_{71}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{71}$
Dimension:	$2$
L-polynomial:	$( 1 + 4 x + 71 x^{2} )^{2}$
	$1 + 8 x + 158 x^{2} + 568 x^{3} + 5041 x^{4}$
Frobenius angles:	$\pm0.576280895962$, $\pm0.576280895962$
Angle rank:	$1$ (numerical)
Jacobians:	$46$
Cyclic group of points:	no
Non-cyclic primes:	$2, 19$

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5776$	$26708224$	$127537551376$	$645459145068544$	$3255529076565442576$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$80$	$5294$	$356336$	$25400094$	$1804387600$	$128100473678$	$9095108163760$	$645753565751614$	$45848501432044496$	$3255243545705583854$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 46 curves (of which all are hyperelliptic):

• $y^2=16 x^6+29 x^5+41 x^4+8 x^3+3 x^2+62 x+55$
• $y^2=18 x^6+15 x^5+32 x^4+54 x^3+45 x^2+54 x+29$
• $y^2=11 x^6+13 x^5+44 x^4+28 x^3+21 x^2+21 x+45$
• $y^2=34 x^6+22 x^5+65 x^4+57 x^3+62 x^2+33 x+4$
• $y^2=33 x^6+40 x^5+14 x^3+29 x^2+37 x+4$
• $y^2=44 x^6+4 x^5+59 x^4+5 x^3+59 x^2+4 x+44$
• $y^2=55 x^6+50 x^5+27 x^4+6 x^3+43 x^2+66 x+52$
• $y^2=8 x^6+53 x^5+57 x^4+65 x^3+61 x^2+66 x+39$
• $y^2=15 x^6+28 x^5+x^4+4 x^3+49 x^2+52 x+47$
• $y^2=7 x^6+44 x^4+44 x^2+7$
• $y^2=65 x^6+33 x^5+39 x^4+55 x^3+23 x^2+5 x+39$
• $y^2=62 x^6+31 x^5+43 x^4+42 x^3+21 x^2+19 x+53$
• $y^2=55 x^6+63 x^5+15 x^4+40 x^3+15 x^2+63 x+55$
• $y^2=4 x^6+17 x^5+30 x^4+49 x^3+30 x^2+17 x+4$
• $y^2=31 x^6+66 x^4+66 x^2+31$
• $y^2=6 x^6+27 x^5+4 x^4+42 x^3+24 x^2+49 x+18$
• $y^2=41 x^6+31 x^5+11 x^4+49 x^3+62 x^2+27 x+63$
• $y^2=56 x^6+23 x^5+64 x^4+3 x^3+27 x^2+58 x+68$
• $y^2=47 x^6+36 x^5+12 x^4+9 x^2+52 x+60$
• $y^2=43 x^6+36 x^5+36 x^4+47 x^3+43 x^2+65 x+54$
• and

• $y^2=51 x^6+14 x^4+14 x^2+51$
• $y^2=61 x^6+69 x^5+3 x^4+19 x^3+69 x^2+21 x+53$
• $y^2=62 x^6+3 x^5+33 x^4+40 x^3+65 x^2+21 x+16$
• $y^2=29 x^6+4 x^5+52 x^4+6 x^3+54 x^2+43 x+21$
• $y^2=34 x^6+66 x^5+29 x^4+47 x^3+36 x^2+58 x+33$
• $y^2=35 x^6+40 x^5+51 x^4+43 x^3+51 x^2+40 x+35$
• $y^2=35 x^6+16 x^5+13 x^4+59 x^3+55 x^2+56 x+28$
• $y^2=26 x^6+16 x^5+19 x^4+30 x^3+58 x^2+x+47$
• $y^2=26 x^6+39 x^5+51 x^4+35 x^3+54 x^2+44 x+12$
• $y^2=40 x^6+2 x^5+70 x^4+42 x^3+67 x^2+26 x+20$
• $y^2=37 x^6+45 x^5+59 x^4+x^3+42 x^2+62 x+9$
• $y^2=2 x^6+46 x^5+40 x^4+30 x^3+47 x^2+23 x+42$
• $y^2=49 x^6+5 x^5+69 x^4+45 x^3+21 x^2+28 x+42$
• $y^2=17 x^6+49 x^5+37 x^4+24 x^3+69 x+36$
• $y^2=22 x^6+8 x^5+33 x^4+27 x^3+31 x^2+19 x+40$
• $y^2=4 x^6+62 x^5+52 x^4+33 x^3+22 x^2+69 x+21$
• $y^2=55 x^6+43 x^5+69 x^4+59 x^3+11 x^2+48 x+70$
• $y^2=70 x^6+20 x^5+15 x^4+16 x^3+25 x^2+69 x+7$
• $y^2=27 x^6+23 x^5+7 x^4+33 x^3+14 x^2+68 x+12$
• $y^2=57 x^6+56 x^5+36 x^4+16 x^3+63 x^2+54 x+13$
• $y^2=52 x^6+63 x^5+34 x^4+7 x^3+40 x^2+56 x+51$
• $y^2=24 x^6+61 x^5+21 x^4+38 x^3+18 x^2+65 x+25$
• $y^2=6 x^6+6 x^5+22 x^4+28 x^3+58 x^2+10 x+39$
• $y^2=15 x^6+22 x^5+46 x^4+67 x^3+17 x^2+33 x+24$
• $y^2=30 x^6+29 x^5+57 x^4+54 x^3+37 x^2+18 x+60$
• $y^2=5 x^6+47 x^5+9 x^4+50 x^3+47 x^2+45 x+47$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{71}$.

Endomorphism algebra over $\F_{71}$

The isogeny class factors as 1.71.e^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-67}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.71.ai_gc	$2$	(not in LMFDB)
2.71.a_ew	$2$	(not in LMFDB)
2.71.ae_acd	$3$	(not in LMFDB)