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

• Label: 2.71.ae_fq
• 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:	$( 1 - 2 x + 71 x^{2} )^{2}$
	$1 - 4 x + 146 x^{2} - 284 x^{3} + 5041 x^{4}$
Frobenius angles:	$\pm0.462134322676$, $\pm0.462134322676$
Angle rank:	$1$ (numerical)
Jacobians:	$50$
Cyclic group of points:	no
Non-cyclic primes:	$2, 5, 7$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4900$	$26832400$	$128400388900$	$645298183398400$	$3255071787033062500$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$68$	$5318$	$358748$	$25393758$	$1804134148$	$128101366118$	$9095129082268$	$645753472257598$	$45848499966877508$	$3255243553694897798$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 50 curves (of which all are hyperelliptic):

• $y^2=52 x^6+50 x^5+48 x^4+29 x^3+57 x^2+40 x+21$
• $y^2=9 x^6+15 x^5+33 x^4+20 x^3+50 x^2+28 x+44$
• $y^2=11 x^6+32 x^5+11 x^4+39 x^3+57 x^2+52 x+36$
• $y^2=22 x^6+51 x^5+70 x^4+40 x^3+15 x^2+65 x+62$
• $y^2=53 x^6+31 x^5+9 x^4+56 x^3+64 x^2+39 x+63$
• $y^2=25 x^6+45 x^5+9 x^4+44 x^3+29 x^2+57 x+10$
• $y^2=20 x^6+4 x^5+36 x^4+63 x^3+69 x^2+6 x+70$
• $y^2=14 x^6+38 x^5+23 x^4+60 x^3+4 x^2+13 x+35$
• $y^2=54 x^6+43 x^5+37 x^4+40 x^3+56 x^2+42 x+9$
• $y^2=64 x^6+13 x^5+41 x^4+13 x^3+10 x^2+14 x+64$
• $y^2=52 x^6+40 x^5+55 x^4+40 x^3+55 x^2+40 x+52$
• $y^2=21 x^6+43 x^5+33 x^4+70 x^3+37 x^2+9$
• $y^2=64 x^6+64 x^5+66 x^4+62 x^3+66 x^2+64 x+64$
• $y^2=57 x^6+9 x^5+63 x^4+65 x^3+49 x^2+41 x+50$
• $y^2=48 x^6+31 x^5+64 x^4+9 x^3+53 x^2+35 x+36$
• $y^2=31 x^6+59 x^5+68 x^4+17 x^3+14 x^2+16 x+24$
• $y^2=69 x^6+14 x^5+31 x^4+37 x^3+67 x^2+47 x+47$
• $y^2=26 x^6+20 x^5+5 x^4+12 x^3+21 x^2+58 x+32$
• $y^2=24 x^6+18 x^5+50 x^4+34 x^3+57 x^2+29 x+62$
• $y^2=6 x^6+60 x^5+39 x^4+36 x^3+39 x^2+60 x+6$
• and

• $y^2=15 x^6+5 x^5+5 x^4+55 x^3+8 x^2+27 x+16$
• $y^2=57 x^6+57 x^5+22 x^4+43 x^3+8 x^2+11 x+70$
• $y^2=9 x^5+70 x^4+25 x^3+44 x^2+68 x+4$
• $y^2=69 x^6+27 x^5+43 x^4+68 x^3+39 x^2+14 x+22$
• $y^2=65 x^6+23 x^5+31 x^4+51 x^3+31 x^2+52 x+29$
• $y^2=52 x^6+40 x^5+51 x^4+65 x^3+51 x^2+32 x+53$
• $y^2=13 x^6+5 x^5+66 x^4+5 x^3+31 x^2+9 x+28$
• $y^2=70 x^6+18 x^5+47 x^4+x^3+47 x^2+18 x+70$
• $y^2=36 x^6+47 x^5+51 x^4+6 x^3+61 x+26$
• $y^2=35 x^6+50 x^5+7 x^4+54 x^3+27 x^2+56 x+64$
• $y^2=27 x^6+23 x^5+63 x^4+42 x^3+47 x^2+65 x+19$
• $y^2=50 x^6+68 x^5+44 x^4+62 x^3+58 x^2+46 x+51$
• $y^2=4 x^6+12 x^5+19 x^4+59 x^3+13 x^2+35 x+55$
• $y^2=30 x^6+48 x^4+48 x^2+30$
• $y^2=12 x^6+67 x^5+55 x^4+70 x^3+64 x^2+4 x+66$
• $y^2=45 x^6+38 x^5+70 x^4+62 x^3+35 x^2+2 x+54$
• $y^2=48 x^6+35 x^5+17 x^4+66 x^3+47 x^2+47 x+5$
• $y^2=24 x^6+7 x^4+7 x^2+24$
• $y^2=46 x^6+51 x^5+35 x^4+39 x^3+33 x^2+37 x+59$
• $y^2=3 x^6+60 x^5+29 x^4+53 x^3+29 x^2+60 x+3$
• $y^2=69 x^6+47 x^5+5 x^4+59 x^3+5 x^2+64 x+46$
• $y^2=62 x^6+7 x^5+44 x^4+42 x^2+23 x+48$
• $y^2=58 x^6+28 x^5+x^4+6 x^3+45 x^2+42 x+10$
• $y^2=21 x^6+66 x^5+12 x^4+56 x^3+10 x^2+45 x+30$
• $y^2=38 x^6+41 x^5+26 x^4+10 x^3+9 x^2+42 x+65$
• $y^2=34 x^6+18 x^5+62 x^3+40 x^2+53 x+68$
• $y^2=52 x^6+32 x^5+52 x^4+27 x^3+52 x^2+32 x+52$
• $y^2=55 x^6+60 x^5+37 x^4+41 x^3+39 x^2+50 x+52$
• $y^2=46 x^6+23 x^5+57 x^4+43 x^3+50 x^2+51 x+56$
• $y^2=9 x^6+70 x^5+46 x^4+10 x^3+54 x^2+63 x+37$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{71}$

The isogeny class factors as 1.71.ac^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-70}) \)$)$

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_fi	$2$	(not in LMFDB)
2.71.e_fq	$2$	(not in LMFDB)
2.71.c_acp	$3$	(not in LMFDB)