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

• Label: 2.71.y_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:	$\pm0.752241693036$, $\pm0.752241693036$
Angle rank:	$1$ (numerical)
Jacobians:	$56$
Cyclic group of points:	no
Non-cyclic primes:	$2, 3, 7$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$7056$	$25401600$	$127508983056$	$646265881497600$	$3255019299989177616$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$96$	$5038$	$356256$	$25431838$	$1804105056$	$128100344398$	$9095128257696$	$645753429760318$	$45848501361223776$	$3255243550502050798$

Jacobians and polarizations

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

• $y^2=37 x^6+65 x^5+44 x^4+70 x^3+70 x^2+52 x+60$
• $y^2=22 x^6+10 x^5+51 x^4+43 x^3+6 x^2+25 x+57$
• $y^2=47 x^6+53 x^5+42 x^4+11 x^3+56 x^2+39 x+22$
• $y^2=65 x^6+57 x^5+24 x^4+69 x^3+24 x^2+57 x+65$
• $y^2=52 x^6+11 x^5+11 x^4+23 x^3+17 x^2+55 x+12$
• $y^2=15 x^6+51 x^5+66 x^4+39 x^3+7 x^2+46 x+27$
• $y^2=19 x^6+55 x^4+55 x^2+19$
• $y^2=9 x^6+15 x^5+29 x^4+18 x^3+13 x^2+55 x+45$
• $y^2=69 x^6+26 x^5+52 x^4+46 x^3+47 x^2+11 x+34$
• $y^2=37 x^6+36 x^4+36 x^2+37$
• $y^2=12 x^6+52 x^5+2 x^4+24 x^3+40 x^2+68 x+8$
• $y^2=46 x^6+4 x^5+43 x^4+57 x^3+9 x^2+37 x+42$
• $y^2=6 x^6+65 x^5+12 x^4+49 x^3+12 x^2+65 x+6$
• $y^2=4 x^6+63 x^5+21 x^4+47 x^3+12 x^2+34 x+48$
• $y^2=7 x^6+38 x^5+30 x^4+64 x^3+49 x^2+40 x+22$
• $y^2=46 x^6+4 x^5+38 x^4+67 x^3+15 x^2+2 x+67$
• $y^2=36 x^6+38 x^5+22 x^4+69 x^3+22 x^2+38 x+36$
• $y^2=29 x^6+8 x^5+11 x^4+13 x^3+11 x^2+8 x+29$
• $y^2=4 x^6+x^5+15 x^4+60 x^3+4 x^2+25 x+3$
• $y^2=56 x^6+64 x^5+55 x^4+30 x^3+7 x^2+30 x+34$
• and

• $y^2=64 x^6+43 x^5+22 x^4+36 x^3+28 x^2+x+10$
• $y^2=17 x^6+32 x^5+51 x^4+34 x^3+24 x^2+14 x+13$
• $y^2=33 x^6+33 x^4+33 x^2+33$
• $y^2=64 x^6+43 x^5+64 x^4+x^2+40 x+29$
• $y^2=10 x^6+28 x^5+43 x^4+37 x^3+43 x^2+28 x+10$
• $y^2=10 x^6+14 x^5+20 x^4+62 x^3+20 x^2+14 x+10$
• $y^2=2 x^6+26 x^5+17 x^4+10 x^3+17 x^2+26 x+2$
• $y^2=64 x^6+16 x^5+3 x^4+46 x^3+25 x^2+54 x+25$
• $y^2=12 x^6+51 x^4+48 x^3+23 x^2+61 x+26$
• $y^2=40 x^6+4 x^5+9 x^4+34 x^3+64 x^2+63 x+55$
• $y^2=9 x^6+34 x^4+34 x^2+9$
• $y^2=4 x^6+3 x^5+44 x^4+5 x^3+44 x^2+3 x+4$
• $y^2=25 x^6+31 x^5+18 x^4+70 x^3+4 x^2+46 x+62$
• $y^2=57 x^6+15 x^5+45 x^4+32 x^3+58 x^2+57 x+16$
• $y^2=40 x^6+15 x^5+8 x^4+41 x^3+48 x^2+38 x+59$
• $y^2=8 x^6+32 x^5+45 x^4+23 x^3+45 x^2+32 x+8$
• $y^2=47 x^6+65 x^5+31 x^4+14 x^3+42 x^2+69 x+51$
• $y^2=20 x^6+62 x^5+20 x^4+41 x^3+49 x^2+31 x+60$
• $y^2=23 x^6+11 x^5+38 x^4+20 x^3+34 x^2+61 x+27$
• $y^2=58 x^6+50 x^5+x^4+4 x^3+48 x^2+38 x+54$
• $y^2=24 x^6+17 x^5+14 x^4+9 x^3+14 x^2+17 x+24$
• $y^2=38 x^6+7 x^5+39 x^4+12 x^3+55 x^2+55 x+58$
• $y^2=61 x^6+57 x^5+23 x^4+44 x^3+23 x^2+57 x+61$
• $y^2=13 x^6+14 x^5+20 x^4+18 x^3+20 x^2+14 x+13$
• $y^2=6 x^6+x^5+9 x^4+12 x^3+9 x^2+x+6$
• $y^2=47 x^6+4 x^5+67 x^4+25 x^3+67 x^2+4 x+47$
• $y^2=7 x^6+35 x^5+68 x^4+12 x^3+68 x^2+35 x+7$
• $y^2=3 x^6+x^4+32 x^3+x^2+3$
• $y^2=41 x^6+45 x^5+55 x^4+64 x^3+55 x^2+45 x+41$
• $y^2=7 x^6+65 x^5+10 x^4+50 x^3+43 x^2+41 x+44$
• $y^2=32 x^6+55 x^5+42 x^4+61 x^3+42 x^2+55 x+32$
• $y^2=19 x^6+35 x^5+27 x^4+52 x^3+27 x^2+35 x+19$
• $y^2=23 x^6+64 x^5+56 x^4+61 x^3+37 x^2+x+60$
• $y^2=49 x^6+47 x^4+47 x^2+49$
• $y^2=60 x^6+40 x^4+40 x^2+60$
• $y^2=34 x^6+9 x^5+26 x^4+36 x^3+67 x^2+8 x+20$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{71}$

The isogeny class factors as 1.71.m^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.ay_la	$2$	(not in LMFDB)
2.71.a_ac	$2$	(not in LMFDB)
2.71.am_cv	$3$	(not in LMFDB)