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

• Label: 2.71.i_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:	$\pm0.549034599758$, $\pm0.604111204402$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5774$	$26687428$	$127554640766$	$645478039315088$	$3255503238755200894$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$80$	$5290$	$356384$	$25400838$	$1804373280$	$128100441898$	$9095110303856$	$645753558351294$	$45848501223640976$	$3255243547540902090$

Jacobians and polarizations

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

• $y^2=24 x^6+40 x^5+57 x^4+64 x^3+41 x^2+66 x+38$
• $y^2=68 x^6+16 x^5+23 x^4+38 x^3+20 x^2+9 x+16$
• $y^2=50 x^6+43 x^5+66 x^4+23 x^3+28 x^2+68 x+17$
• $y^2=24 x^6+54 x^5+5 x^4+38 x^3+51 x^2+41 x+67$
• $y^2=38 x^6+9 x^5+62 x^4+3 x^3+55 x^2+16 x$
• $y^2=34 x^6+45 x^5+21 x^4+36 x^3+61 x^2+38 x+22$
• $y^2=27 x^6+31 x^5+48 x^4+32 x^3+67 x^2+4 x+51$
• $y^2=37 x^6+17 x^5+51 x^4+68 x^3+27 x^2+6 x+44$
• $y^2=36 x^6+15 x^5+42 x^4+66 x^3+x^2+20 x+34$
• $y^2=9 x^6+20 x^5+14 x^4+8 x^3+18 x^2+16 x+6$
• $y^2=44 x^6+47 x^5+2 x^4+47 x^3+23 x^2+70 x+70$
• $y^2=33 x^6+40 x^4+21 x^3+46 x^2+69 x+59$
• $y^2=70 x^6+40 x^5+41 x^4+59 x^3+52 x^2+63 x+51$
• $y^2=40 x^6+6 x^5+24 x^4+55 x^3+57 x^2+14 x+31$
• $y^2=37 x^6+48 x^4+70 x^3+55 x^2+13 x+43$
• $y^2=53 x^6+54 x^5+41 x^4+9 x^3+54 x^2+44 x+69$
• $y^2=70 x^6+64 x^5+63 x^4+3 x^3+22 x^2+6 x+47$
• $y^2=45 x^6+9 x^5+5 x^4+14 x^3+22 x^2+38 x+68$
• $y^2=55 x^6+56 x^5+41 x^4+9 x^3+40 x^2+22 x+9$
• $y^2=x^6+68 x^5+59 x^4+38 x^3+40 x^2+11 x+37$
• and

• $y^2=51 x^6+20 x^5+31 x^4+6 x^3+15 x^2+55 x+6$
• $y^2=6 x^6+16 x^5+43 x^4+37 x^3+65 x^2+23 x+19$
• $y^2=46 x^6+16 x^5+38 x^4+2 x^3+57 x^2+33 x+45$
• $y^2=52 x^6+38 x^5+51 x^4+5 x^3+30 x^2+54 x+54$
• $y^2=9 x^6+26 x^5+26 x^4+x^3+65 x^2+49 x+31$
• $y^2=57 x^6+13 x^5+18 x^4+63 x^3+16 x^2+31 x+20$
• $y^2=23 x^6+28 x^5+45 x^4+47 x^3+32 x^2+44 x+35$
• $y^2=9 x^6+68 x^5+10 x^4+34 x^3+40 x^2+53 x+55$
• $y^2=54 x^6+x^5+44 x^4+34 x^3+38 x^2+4 x+53$
• $y^2=69 x^6+2 x^5+55 x^4+28 x^3+20 x^2+29 x+56$
• $y^2=6 x^6+21 x^5+20 x^4+65 x^3+49 x^2+36 x+11$
• $y^2=15 x^6+54 x^5+62 x^4+24 x^3+12 x^2+15 x+58$
• $y^2=51 x^6+8 x^5+34 x^4+2 x^3+42 x^2+x+5$
• $y^2=61 x^6+48 x^5+36 x^4+55 x^3+52 x^2+27 x+15$

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.ai_ga	$2$	(not in LMFDB)