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

• Label: 2.71.j_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:	$( 1 + 3 x + 71 x^{2} )( 1 + 6 x + 71 x^{2} )$
	$1 + 9 x + 160 x^{2} + 639 x^{3} + 5041 x^{4}$
Frobenius angles:	$\pm0.556968285566$, $\pm0.615871442562$
Angle rank:	$2$ (numerical)
Jacobians:	$40$
Cyclic group of points:	no
Non-cyclic primes:	$3, 5$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5850$	$26640900$	$127502505000$	$645530959101600$	$3255511666751808750$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$81$	$5281$	$356238$	$25402921$	$1804377951$	$128100213178$	$9095111056161$	$645753573694321$	$45848501088886818$	$3255243547057117801$

Jacobians and polarizations

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

• $y^2=67 x^6+42 x^5+56 x^4+43 x^3+4 x^2+11 x+21$
• $y^2=38 x^6+57 x^5+40 x^4+7 x^3+67 x^2+13 x+36$
• $y^2=38 x^6+19 x^5+21 x^4+17 x^3+20 x^2+61 x$
• $y^2=3 x^6+60 x^5+39 x^4+30 x^3+51 x^2+34 x+10$
• $y^2=9 x^6+8 x^5+27 x^4+30 x^3+28 x^2+62 x+32$
• $y^2=43 x^6+58 x^5+26 x^4+60 x^3+70 x^2+2 x+67$
• $y^2=13 x^6+5 x^5+14 x^4+54 x^3+47 x^2+58 x+39$
• $y^2=7 x^6+17 x^5+53 x^4+23 x^3+56 x^2+63 x+52$
• $y^2=68 x^6+57 x^5+4 x^4+11 x^3+48 x^2+24 x+60$
• $y^2=44 x^6+38 x^5+62 x^4+67 x^3+61 x^2+35 x+15$
• $y^2=14 x^6+61 x^5+56 x^4+42 x^3+45 x^2+3 x+7$
• $y^2=69 x^6+60 x^5+65 x^4+34 x^3+62 x^2+38 x+50$
• $y^2=7 x^6+46 x^5+24 x^4+46 x^3+27 x^2+14 x+31$
• $y^2=12 x^6+41 x^5+22 x^4+21 x^3+43 x^2+43 x+32$
• $y^2=9 x^6+8 x^5+3 x^4+53 x^3+60 x^2+64$
• $y^2=63 x^6+12 x^5+11 x^4+52 x^3+47 x^2+70 x+45$
• $y^2=40 x^6+22 x^5+43 x^4+70 x^3+48 x^2+25 x+41$
• $y^2=29 x^6+10 x^5+61 x^4+9 x^3+24 x^2+58 x+23$
• $y^2=12 x^6+29 x^5+47 x^4+63 x^3+41 x^2+15 x$
• $y^2=31 x^6+54 x^5+38 x^4+54 x^3+59 x^2+29 x+19$
• and

• $y^2=62 x^6+70 x^5+15 x^4+54 x^3+29 x^2+61 x+19$
• $y^2=5 x^5+24 x^4+28 x^3+x^2+24 x+5$
• $y^2=68 x^6+9 x^5+51 x^4+44 x^3+9 x^2+10 x+38$
• $y^2=49 x^6+23 x^5+25 x^4+22 x^3+10 x^2+18 x+66$
• $y^2=56 x^6+27 x^5+19 x^4+22 x^3+x^2+42 x+30$
• $y^2=6 x^6+7 x^5+69 x^4+54 x^3+38 x^2+70 x+66$
• $y^2=65 x^6+35 x^5+38 x^4+39 x^3+61 x^2+36 x+39$
• $y^2=60 x^6+63 x^5+61 x^4+51 x^3+27 x^2+19 x+6$
• $y^2=47 x^6+43 x^5+70 x^4+66 x^3+12 x^2+34 x+69$
• $y^2=29 x^6+25 x^5+70 x^4+69 x^3+23 x^2+35 x+48$
• $y^2=43 x^6+26 x^5+52 x^4+4 x^3+6 x^2+24 x+17$
• $y^2=31 x^6+20 x^5+48 x^4+48 x^3+15 x^2+68 x+60$
• $y^2=51 x^6+32 x^5+26 x^4+14 x^3+43 x^2+23 x+4$
• $y^2=19 x^6+36 x^5+3 x^4+45 x^3+38 x^2+18 x+15$
• $y^2=17 x^6+3 x^5+67 x^4+48 x^3+27 x^2+41 x+43$
• $y^2=59 x^6+60 x^5+22 x^4+19 x^3+x^2+36 x+66$
• $y^2=37 x^6+2 x^5+20 x^4+60 x^3+40 x^2+70 x+28$
• $y^2=64 x^6+38 x^5+9 x^3+16 x^2+62 x+24$
• $y^2=25 x^6+69 x^5+6 x^4+3 x^3+49 x^2+52 x+47$
• $y^2=5 x^6+42 x^5+37 x^4+28 x^3+31 x^2+65 x+36$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{71}$

The isogeny class factors as 1.71.d $\times$ 1.71.g 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.d : \(\Q(\sqrt{-11}) \).
• 1.71.g : \(\Q(\sqrt{-62}) \).

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