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

• Label: 2.71.ay_la
• 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:	$( 1 - 12 x + 71 x^{2} )^{2}$
	$1 - 24 x + 286 x^{2} - 1704 x^{3} + 5041 x^{4}$
Frobenius angles:	$\pm0.247758306964$, $\pm0.247758306964$
Angle rank:	$1$ (numerical)
Jacobians:	$56$
Cyclic group of points:	no
Non-cyclic primes:	$2, 3, 5$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3600$	$25401600$	$128694387600$	$646265881497600$	$3255467816972250000$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$48$	$5038$	$359568$	$25431838$	$1804353648$	$128100344398$	$9095112059088$	$645753429760318$	$45848500075674288$	$3255243550502050798$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 56 curves (of which all are hyperelliptic):

• $y^2=56 x^6+60 x^5+57 x^4+10 x^3+10 x^2+48 x+39$
• $y^2=64 x^6+42 x^5+58 x^4+67 x^3+11 x^2+34 x+69$
• $y^2=27 x^6+38 x^5+6 x^4+32 x^3+8 x^2+36 x+64$
• $y^2=29 x^6+44 x^5+26 x^4+57 x^3+26 x^2+44 x+29$
• $y^2=9 x^6+6 x^5+6 x^4+19 x^3+48 x^2+30 x+13$
• $y^2=63 x^6+58 x^5+50 x^4+36 x^3+x^2+37 x+14$
• $y^2=62 x^6+30 x^4+30 x^2+62$
• $y^2=52 x^6+63 x^5+65 x^4+33 x^3+12 x^2+18 x+47$
• $y^2=20 x^6+24 x^5+48 x^4+37 x^3+27 x^2+32 x+15$
• $y^2=46 x^6+39 x^4+39 x^2+46$
• $y^2=22 x^6+48 x^5+51 x^4+44 x^3+26 x^2+30 x+62$
• $y^2=38 x^6+28 x^5+17 x^4+44 x^3+63 x^2+46 x+10$
• $y^2=42 x^6+29 x^5+13 x^4+59 x^3+13 x^2+29 x+42$
• $y^2=28 x^6+15 x^5+5 x^4+45 x^3+13 x^2+25 x+52$
• $y^2=49 x^6+53 x^5+68 x^4+22 x^3+59 x^2+67 x+12$
• $y^2=37 x^6+31 x^5+46 x^4+40 x^3+63 x^2+51 x+40$
• $y^2=39 x^6+53 x^5+12 x^4+57 x^3+12 x^2+53 x+39$
• $y^2=61 x^6+56 x^5+6 x^4+20 x^3+6 x^2+56 x+61$
• $y^2=28 x^6+7 x^5+34 x^4+65 x^3+28 x^2+33 x+21$
• $y^2=8 x^6+70 x^5+18 x^4+55 x^3+x^2+55 x+15$
• and

• $y^2=70 x^6+67 x^5+64 x^4+66 x^3+4 x^2+61 x+42$
• $y^2=48 x^6+11 x^5+2 x^4+25 x^3+26 x^2+27 x+20$
• $y^2=25 x^6+25 x^4+25 x^2+25$
• $y^2=70 x^6+67 x^5+70 x^4+61 x^2+26 x+65$
• $y^2=70 x^6+54 x^5+17 x^4+46 x^3+17 x^2+54 x+70$
• $y^2=70 x^6+27 x^5+69 x^4+8 x^3+69 x^2+27 x+70$
• $y^2=14 x^6+40 x^5+48 x^4+70 x^3+48 x^2+40 x+14$
• $y^2=22 x^6+41 x^5+21 x^4+38 x^3+33 x^2+23 x+33$
• $y^2=22 x^6+58 x^4+17 x^3+54 x^2+29 x+24$
• $y^2=26 x^6+31 x^5+52 x^4+15 x^3+70 x^2+9 x+18$
• $y^2=63 x^6+25 x^4+25 x^2+63$
• $y^2=28 x^6+21 x^5+24 x^4+35 x^3+24 x^2+21 x+28$
• $y^2=34 x^6+45 x^5+33 x^4+10 x^3+31 x^2+37 x+19$
• $y^2=44 x^6+34 x^5+31 x^4+11 x^3+51 x^2+44 x+41$
• $y^2=67 x^6+34 x^5+56 x^4+3 x^3+52 x^2+53 x+58$
• $y^2=62 x^6+35 x^5+47 x^4+54 x^3+47 x^2+35 x+62$
• $y^2=27 x^6+60 x^5+45 x^4+2 x^3+6 x^2+20 x+58$
• $y^2=13 x^6+19 x^5+13 x^4+16 x^3+7 x^2+45 x+39$
• $y^2=54 x^6+32 x^5+46 x^4+13 x^3+15 x^2+29 x+14$
• $y^2=59 x^6+68 x^5+61 x^4+31 x^3+17 x^2+46 x+28$
• $y^2=26 x^6+48 x^5+27 x^4+63 x^3+27 x^2+48 x+26$
• $y^2=46 x^6+x^5+36 x^4+22 x^3+18 x^2+18 x+59$
• $y^2=29 x^6+69 x^5+54 x^4+57 x^3+54 x^2+69 x+29$
• $y^2=12 x^6+2 x^5+13 x^4+33 x^3+13 x^2+2 x+12$
• $y^2=11 x^6+61 x^5+52 x^4+22 x^3+52 x^2+61 x+11$
• $y^2=27 x^6+31 x^5+40 x^4+34 x^3+40 x^2+31 x+27$
• $y^2=49 x^6+32 x^5+50 x^4+13 x^3+50 x^2+32 x+49$
• $y^2=21 x^6+7 x^4+11 x^3+7 x^2+21$
• $y^2=16 x^6+47 x^5+18 x^4+70 x^3+18 x^2+47 x+16$
• $y^2=49 x^6+29 x^5+70 x^4+66 x^3+17 x^2+3 x+24$
• $y^2=35 x^6+18 x^5+6 x^4+29 x^3+6 x^2+18 x+35$
• $y^2=62 x^6+32 x^5+47 x^4+9 x^3+47 x^2+32 x+62$
• $y^2=54 x^6+70 x^5+8 x^4+29 x^3+56 x^2+61 x+39$
• $y^2=59 x^6+45 x^4+45 x^2+59$
• $y^2=39 x^6+26 x^4+26 x^2+39$
• $y^2=25 x^6+63 x^5+40 x^4+39 x^3+43 x^2+56 x+69$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{71}$

The isogeny class factors as 1.71.am^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-35}) \)$)$

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.a_ac	$2$	(not in LMFDB)
2.71.y_la	$2$	(not in LMFDB)
2.71.m_cv	$3$	(not in LMFDB)