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

• Label: 2.71.aj_ge
• 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:	$( 1 - 6 x + 71 x^{2} )( 1 - 3 x + 71 x^{2} )$
	$1 - 9 x + 160 x^{2} - 639 x^{3} + 5041 x^{4}$
Frobenius angles:	$\pm0.384128557438$, $\pm0.443031714434$
Angle rank:	$2$ (numerical)
Jacobians:	$40$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4554$	$26640900$	$128700794376$	$645530959101600$	$3254975453396853054$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$63$	$5281$	$359586$	$25402921$	$1804080753$	$128100213178$	$9095129260623$	$645753573694321$	$45848500348011246$	$3255243547057117801$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 40 curves (of which all are hyperelliptic):

• $y^2=40 x^6+6 x^5+8 x^4+67 x^3+31 x^2+32 x+3$
• $y^2=46 x^6+69 x^5+26 x^4+x^3+40 x^2+12 x+66$
• $y^2=46 x^6+23 x^5+3 x^4+43 x^3+13 x^2+29 x$
• $y^2=21 x^6+65 x^5+60 x^4+68 x^3+2 x^2+25 x+70$
• $y^2=63 x^6+56 x^5+47 x^4+68 x^3+54 x^2+8 x+11$
• $y^2=17 x^6+51 x^5+40 x^4+65 x^3+64 x^2+14 x+43$
• $y^2=20 x^6+35 x^5+27 x^4+23 x^3+45 x^2+51 x+60$
• $y^2=49 x^6+48 x^5+16 x^4+19 x^3+37 x^2+15 x+9$
• $y^2=30 x^6+69 x^5+31 x^4+32 x^3+17 x^2+44 x+39$
• $y^2=24 x^6+53 x^5+8 x^4+43 x^3+x^2+32 x+34$
• $y^2=2 x^6+29 x^5+8 x^4+6 x^3+47 x^2+41 x+1$
• $y^2=20 x^6+39 x^5+60 x^4+15 x^3+19 x^2+46 x+68$
• $y^2=49 x^6+38 x^5+26 x^4+38 x^3+47 x^2+27 x+4$
• $y^2=13 x^6+3 x^5+12 x^4+5 x^3+17 x^2+17 x+11$
• $y^2=63 x^6+56 x^5+21 x^4+16 x^3+65 x^2+22$
• $y^2=15 x^6+13 x^5+6 x^4+9 x^3+45 x^2+64 x+31$
• $y^2=26 x^6+64 x^5+67 x^4+10 x^3+17 x^2+34 x+16$
• $y^2=61 x^6+70 x^5+x^4+63 x^3+26 x^2+51 x+19$
• $y^2=13 x^6+61 x^5+45 x^4+15 x^3+3 x^2+34 x$
• $y^2=45 x^6+28 x^5+46 x^4+28 x^3+49 x^2+65 x+23$
• and

• $y^2=19 x^6+10 x^5+63 x^4+28 x^3+65 x^2+29 x+23$
• $y^2=21 x^5+44 x^4+4 x^3+61 x^2+44 x+21$
• $y^2=30 x^6+52 x^5+58 x^4+57 x^3+52 x^2+42 x+46$
• $y^2=59 x^6+19 x^5+33 x^4+12 x^3+70 x^2+55 x+36$
• $y^2=8 x^6+14 x^5+23 x^4+64 x^3+61 x^2+6 x+55$
• $y^2=42 x^6+49 x^5+57 x^4+23 x^3+53 x^2+64 x+36$
• $y^2=60 x^6+5 x^5+46 x^4+36 x^3+29 x^2+66 x+36$
• $y^2=65 x^6+15 x^5+x^4+2 x^3+47 x^2+62 x+42$
• $y^2=45 x^6+17 x^5+64 x^4+36 x^3+13 x^2+25 x+57$
• $y^2=61 x^6+33 x^5+64 x^4+57 x^3+19 x^2+32 x+52$
• $y^2=67 x^6+24 x^5+48 x^4+31 x^3+11 x^2+44 x+43$
• $y^2=4 x^6+69 x^5+52 x^4+52 x^3+34 x^2+50 x+65$
• $y^2=2 x^6+11 x^5+40 x^4+27 x^3+17 x^2+19 x+28$
• $y^2=23 x^6+66 x^5+41 x^4+47 x^3+46 x^2+33 x+63$
• $y^2=43 x^6+41 x^5+40 x^4+17 x^3+14 x^2+16 x+67$
• $y^2=49 x^6+39 x^5+64 x^4+23 x^3+61 x^2+66 x+50$
• $y^2=56 x^6+51 x^5+13 x^4+39 x^3+26 x^2+10 x+4$
• $y^2=70 x^6+46 x^5+52 x^3+53 x^2+19 x+44$
• $y^2=33 x^6+57 x^5+42 x^4+21 x^3+59 x^2+9 x+45$
• $y^2=21 x^6+6 x^5+56 x^4+4 x^3+45 x^2+60 x+66$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{71}$

The isogeny class factors as 1.71.ag $\times$ 1.71.ad and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.71.ag : \(\Q(\sqrt{-62}) \).
• 1.71.ad : \(\Q(\sqrt{-11}) \).

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.ad_eu	$2$	(not in LMFDB)
2.71.d_eu	$2$	(not in LMFDB)
2.71.j_ge	$2$	(not in LMFDB)