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

• Label: 2.71.ai_ae
• 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:	$1 - 8 x - 4 x^{2} - 568 x^{3} + 5041 x^{4}$
Frobenius angles:	$\pm0.0386966968237$, $\pm0.673288635073$
Angle rank:	$2$ (numerical)
Number field:	4.0.55552.1
Galois group:	$D_{4}$
Jacobians:	$36$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4462$	$25049668$	$127274842078$	$645673128262288$	$3255180173236265182$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$64$	$4970$	$355600$	$25408518$	$1804194224$	$128099039978$	$9095119923968$	$645753520391614$	$45848500101198208$	$3255243552166026890$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 36 curves (of which all are hyperelliptic):

• $y^2=49 x^6+69 x^5+33 x^4+42 x^3+43 x^2+52 x+9$
• $y^2=49 x^6+20 x^5+36 x^4+48 x^3+17 x^2+3 x+53$
• $y^2=61 x^6+30 x^5+17 x^4+36 x^3+54 x^2+15 x+65$
• $y^2=14 x^6+17 x^5+4 x^4+50 x^3+17 x^2+47 x+44$
• $y^2=21 x^6+22 x^5+35 x^4+41 x^3+27 x^2+40 x+34$
• $y^2=18 x^6+56 x^5+22 x^4+25 x^3+31 x^2+53 x+3$
• $y^2=23 x^6+10 x^5+57 x^4+30 x^3+50 x^2+65 x+63$
• $y^2=26 x^6+26 x^5+32 x^4+16 x^3+70 x^2+9 x+46$
• $y^2=68 x^6+61 x^5+38 x^4+26 x^3+60 x^2+3 x+56$
• $y^2=48 x^6+27 x^5+42 x^4+9 x^3+33 x^2+56 x+58$
• $y^2=28 x^6+7 x^5+35 x^4+68 x^3+65 x^2+52 x+62$
• $y^2=55 x^6+54 x^5+41 x^4+30 x^3+4 x^2+59 x+46$
• $y^2=51 x^6+49 x^5+33 x^4+23 x^3+34 x^2+19 x+60$
• $y^2=13 x^6+62 x^5+53 x^4+54 x^3+46 x^2+37 x+55$
• $y^2=30 x^6+60 x^5+19 x^4+57 x^3+45 x^2+32 x+64$
• $y^2=62 x^6+58 x^5+50 x^4+69 x^3+51 x^2+10 x+21$
• $y^2=36 x^6+56 x^5+49 x^4+35 x^3+29 x^2+5 x+49$
• $y^2=67 x^6+33 x^5+51 x^4+62 x^3+9 x^2+14 x+33$
• $y^2=18 x^6+31 x^5+54 x^4+37 x^3+45 x^2+7 x+70$
• $y^2=38 x^6+56 x^5+28 x^4+43 x^3+68 x^2+52 x+33$
• and

• $y^2=63 x^6+68 x^5+23 x^4+52 x^3+50 x^2+21 x+49$
• $y^2=5 x^6+34 x^5+64 x^4+10 x^3+33 x^2+5 x+3$
• $y^2=33 x^6+13 x^4+66 x^3+21 x^2+62 x+52$
• $y^2=23 x^6+34 x^5+51 x^4+11 x^3+44 x^2+56 x+59$
• $y^2=40 x^6+24 x^5+68 x^4+6 x^2+38 x+63$
• $y^2=16 x^6+44 x^5+19 x^4+2 x^3+59 x^2+41 x+67$
• $y^2=16 x^6+28 x^5+58 x^4+21 x^3+41 x^2+55 x+3$
• $y^2=29 x^6+23 x^5+24 x^4+64 x^3+22 x^2+36 x+57$
• $y^2=7 x^6+33 x^5+70 x^4+70 x^3+5 x^2+52 x+64$
• $y^2=41 x^6+46 x^5+41 x^4+17 x^3+46 x^2+16 x+67$
• $y^2=62 x^6+28 x^5+22 x^4+24 x^3+15 x^2+41 x+49$
• $y^2=46 x^6+29 x^5+54 x^4+8 x^3+56 x^2+61 x+26$
• $y^2=44 x^6+22 x^5+34 x^3+50 x^2+22 x+5$
• $y^2=40 x^6+17 x^5+27 x^4+47 x^3+22 x^2+56 x+21$
• $y^2=36 x^6+6 x^5+45 x^4+2 x^3+43 x^2+69 x+36$
• $y^2=7 x^6+45 x^5+10 x^4+53 x^3+6 x^2+x+16$

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 4.0.55552.1.

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.i_ae	$2$	(not in LMFDB)