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

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

Invariants

Base field:	$\F_{71}$
Dimension:	$2$
L-polynomial:	$1 + 4 x + 140 x^{2} + 284 x^{3} + 5041 x^{4}$
Frobenius angles:	$\pm0.491508933344$, $\pm0.585051347051$
Angle rank:	$2$ (numerical)
Number field:	\(\Q(\sqrt{-274 +4 \sqrt{6}})\)
Galois group:	$D_{4}$
Jacobians:	$40$
Isomorphism classes:	40
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5470$	$26770180$	$127827472750$	$645375606520720$	$3255372238539256750$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$76$	$5306$	$357148$	$25396806$	$1804300676$	$128100967418$	$9095115515156$	$645753508863166$	$45848500904375308$	$3255243551272915226$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 40 curves (of which all are hyperelliptic):

• $y^2=49 x^6+2 x^5+46 x^4+7 x^3+6 x^2+40 x+30$
• $y^2=12 x^6+58 x^5+66 x^4+18 x^3+51 x^2+52 x+26$
• $y^2=15 x^6+11 x^5+40 x^4+19 x^3+70 x^2+52 x+5$
• $y^2=23 x^6+15 x^5+67 x^4+64 x^3+50 x^2+20 x+3$
• $y^2=26 x^6+49 x^5+12 x^4+47 x^3+7 x^2+20 x+43$
• $y^2=29 x^6+31 x^5+66 x^4+69 x^3+19 x^2+46 x+64$
• $y^2=4 x^6+57 x^5+60 x^4+70 x^3+12 x^2+65 x+61$
• $y^2=13 x^6+20 x^5+50 x^4+52 x^3+17 x^2+52 x+61$
• $y^2=57 x^6+66 x^5+54 x^4+12 x^3+50 x^2+64$
• $y^2=36 x^6+16 x^5+21 x^4+40 x^3+61 x^2+41 x+42$
• $y^2=47 x^6+2 x^5+14 x^4+36 x^3+60 x^2+39 x+69$
• $y^2=52 x^6+14 x^5+52 x^4+67 x^3+28 x^2+69 x+68$
• $y^2=33 x^6+11 x^5+65 x^4+3 x^3+21 x^2+68 x+31$
• $y^2=50 x^6+52 x^5+51 x^4+23 x^3+41 x^2+49 x+44$
• $y^2=37 x^6+52 x^5+26 x^4+x^3+55 x^2+31 x+62$
• $y^2=6 x^6+3 x^5+19 x^4+10 x^3+65 x^2+45 x+45$
• $y^2=11 x^6+8 x^5+4 x^4+25 x^3+39 x^2+7 x+57$
• $y^2=20 x^6+56 x^5+53 x^4+44 x^3+37 x^2+10 x+45$
• $y^2=6 x^6+62 x^5+34 x^4+8 x^3+53 x^2+64 x+22$
• $y^2=57 x^6+49 x^5+64 x^4+34 x^3+45 x^2+14 x+2$
• and

• $y^2=10 x^6+64 x^5+36 x^4+37 x^3+41 x^2+14 x+7$
• $y^2=5 x^6+50 x^5+25 x^4+34 x^3+26 x^2+21 x+25$
• $y^2=25 x^6+33 x^5+58 x^4+42 x^3+33 x^2+54 x+2$
• $y^2=54 x^6+46 x^5+29 x^4+62 x^3+29 x^2+44 x+31$
• $y^2=21 x^6+52 x^5+70 x^4+20 x^3+21 x^2+59 x+29$
• $y^2=60 x^6+66 x^5+11 x^4+5 x^3+58 x^2+53 x+33$
• $y^2=61 x^6+23 x^5+x^4+8 x^3+22 x^2+40 x+13$
• $y^2=48 x^6+64 x^5+14 x^3+51 x^2+45 x+20$
• $y^2=25 x^6+24 x^5+22 x^4+7 x^3+36 x^2+24 x+70$
• $y^2=7 x^6+7 x^5+52 x^4+2 x^3+46 x^2+31 x+16$
• $y^2=28 x^6+32 x^5+26 x^4+45 x^3+21 x^2+22 x+27$
• $y^2=51 x^6+10 x^5+48 x^4+3 x^3+24 x^2+17 x+13$
• $y^2=44 x^6+32 x^5+37 x^4+22 x^3+67 x^2+38 x+53$
• $y^2=2 x^6+17 x^5+24 x^4+57 x^3+69 x^2+28 x+56$
• $y^2=56 x^6+37 x^5+52 x^4+21 x^3+32 x^2+36 x+61$
• $y^2=28 x^6+51 x^5+2 x^4+56 x^3+24 x^2+11 x+23$
• $y^2=12 x^6+5 x^5+52 x^4+62 x^3+33 x^2+14 x+20$
• $y^2=70 x^6+68 x^5+35 x^4+31 x^3+54 x^2+18 x+21$
• $y^2=19 x^6+10 x^5+47 x^4+41 x^3+20 x^2+49 x+64$
• $y^2=57 x^6+17 x^5+40 x^4+18 x^3+46 x^2+61 x+59$

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 \(\Q(\sqrt{-274 +4 \sqrt{6}})\).

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.ae_fk	$2$	(not in LMFDB)