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

• Label: 2.71.ai_ga
• 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:	$1 - 8 x + 156 x^{2} - 568 x^{3} + 5041 x^{4}$
Frobenius angles:	$\pm0.395888795598$, $\pm0.450965400242$
Angle rank:	$2$ (numerical)
Number field:	\(\Q(\sqrt{-266 +8 \sqrt{2}})\)
Galois group:	$D_{4}$
Jacobians:	$34$
Isomorphism classes:	34
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4622$	$26687428$	$128648419838$	$645478039315088$	$3254983880510851582$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$64$	$5290$	$359440$	$25400838$	$1804085424$	$128100441898$	$9095130012928$	$645753558351294$	$45848500213257088$	$3255243547540902090$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 34 curves (of which all are hyperelliptic):

• $y^2=44 x^6+26 x^5+69 x^4+70 x^3+16 x^2+50 x+46$
• $y^2=30 x^6+53 x^5+54 x^4+46 x^3+13 x^2+52 x+53$
• $y^2=68 x^6+67 x^5+50 x^4+54 x^3+4 x^2+30 x+43$
• $y^2=44 x^6+28 x^5+21 x^4+46 x^3+58 x^2+16 x+40$
• $y^2=53 x^6+63 x^5+8 x^4+21 x^3+30 x^2+41 x$
• $y^2=15 x^6+47 x^5+3 x^4+66 x^3+29 x^2+46 x+64$
• $y^2=47 x^6+4 x^5+52 x^4+11 x^3+43 x^2+28 x+2$
• $y^2=46 x^6+48 x^5+2 x^4+50 x^3+47 x^2+42 x+24$
• $y^2=66 x^6+63 x^5+6 x^4+50 x^3+61 x^2+13 x+15$
• $y^2=63 x^6+69 x^5+27 x^4+56 x^3+55 x^2+41 x+42$
• $y^2=57 x^6+27 x^5+51 x^4+27 x^3+54 x^2+10 x+10$
• $y^2=18 x^6+67 x^4+5 x^3+38 x^2+57 x+58$
• $y^2=10 x^6+26 x^5+16 x^4+49 x^3+48 x^2+9 x+58$
• $y^2=67 x^6+42 x^5+26 x^4+30 x^3+44 x^2+27 x+4$
• $y^2=46 x^6+52 x^4+64 x^3+30 x^2+20 x+17$
• $y^2=38 x^6+28 x^5+16 x^4+52 x^3+28 x^2+57 x+20$
• $y^2=64 x^6+22 x^5+15 x^4+21 x^3+12 x^2+42 x+45$
• $y^2=31 x^6+63 x^5+35 x^4+27 x^3+12 x^2+53 x+50$
• $y^2=30 x^6+37 x^5+3 x^4+63 x^3+67 x^2+12 x+63$
• $y^2=7 x^6+50 x^5+58 x^4+53 x^3+67 x^2+6 x+46$
• and

• $y^2=58 x^6+13 x^5+45 x^4+11 x^3+63 x^2+18 x+11$
• $y^2=42 x^6+41 x^5+17 x^4+46 x^3+29 x^2+19 x+62$
• $y^2=37 x^6+53 x^5+46 x^4+51 x^3+69 x^2+25 x+47$
• $y^2=48 x^6+46 x^5+58 x^4+21 x^3+55 x^2+28 x+28$
• $y^2=63 x^6+40 x^5+40 x^4+7 x^3+29 x^2+59 x+4$
• $y^2=44 x^6+20 x^5+55 x^4+15 x^3+41 x^2+4 x+69$
• $y^2=54 x^6+4 x^5+47 x^4+27 x^3+35 x^2+57 x+5$
• $y^2=52 x^6+30 x^5+42 x^4+15 x^3+26 x^2+38 x+18$
• $y^2=28 x^6+61 x^5+57 x^4+15 x^3+46 x^2+31 x+38$
• $y^2=20 x^6+51 x^5+18 x^4+4 x^3+13 x^2+65 x+8$
• $y^2=42 x^6+5 x^5+69 x^4+29 x^3+59 x^2+39 x+6$
• $y^2=63 x^6+28 x^5+19 x^4+44 x^3+22 x^2+63 x+59$
• $y^2=58 x^6+62 x^5+15 x^4+51 x^3+6 x^2+61 x+21$
• $y^2=x^6+52 x^5+39 x^4+30 x^3+9 x^2+47 x+34$

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 \(\Q(\sqrt{-266 +8 \sqrt{2}})\).

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.i_ga	$2$	(not in LMFDB)