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

• Label: 2.71.o_hi
• 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:	$( 1 + 6 x + 71 x^{2} )( 1 + 8 x + 71 x^{2} )$
	$1 + 14 x + 190 x^{2} + 994 x^{3} + 5041 x^{4}$
Frobenius angles:	$\pm0.615871442562$, $\pm0.657448017853$
Angle rank:	$2$ (numerical)
Jacobians:	$64$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6240$	$26357760$	$127295532000$	$645825848279040$	$3255487083005196000$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$86$	$5226$	$355658$	$25414526$	$1804364326$	$128099166858$	$9095118687706$	$645753615576766$	$45848500247202998$	$3255243548677416426$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 64 curves (of which all are hyperelliptic):

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

• $y^2=38 x^6+69 x^5+20 x^4+20 x^3+20 x^2+69 x+38$
• $y^2=43 x^6+24 x^5+39 x^4+43 x^3+53 x^2+32 x+50$
• $y^2=58 x^6+22 x^5+2 x^4+17 x^3+43 x^2+52 x+30$
• $y^2=35 x^6+46 x^5+47 x^4+25 x^3+67 x^2+21 x+59$
• $y^2=57 x^6+55 x^5+37 x^4+34 x^3+37 x^2+55 x+57$
• $y^2=52 x^6+23 x^5+62 x^4+57 x^3+62 x^2+23 x+52$
• $y^2=12 x^6+28 x^5+15 x^4+20 x^3+27 x^2+68 x+20$
• $y^2=54 x^6+18 x^5+23 x^4+44 x^3+23 x^2+18 x+54$
• $y^2=14 x^6+16 x^5+17 x^4+68 x^3+13 x^2+58 x+67$
• $y^2=29 x^6+42 x^5+18 x^4+22 x^3+18 x^2+42 x+29$
• $y^2=19 x^6+55 x^5+31 x^4+43 x^3+46 x^2+47 x+40$
• $y^2=64 x^6+56 x^5+54 x^4+55 x^3+54 x^2+56 x+64$
• $y^2=3 x^6+33 x^5+48 x^4+51 x^3+48 x^2+33 x+3$
• $y^2=36 x^6+17 x^5+45 x^4+23 x^3+45 x^2+17 x+36$
• $y^2=49 x^6+44 x^5+49 x^4+35 x^3+49 x^2+44 x+49$
• $y^2=35 x^6+3 x^5+43 x^4+37 x^3+43 x^2+3 x+35$
• $y^2=32 x^6+36 x^5+13 x^4+61 x^3+13 x^2+36 x+32$
• $y^2=65 x^6+2 x^5+22 x^4+69 x^3+22 x^2+2 x+65$
• $y^2=42 x^6+29 x^5+29 x^4+36 x^3+29 x^2+29 x+42$
• $y^2=40 x^6+4 x^5+58 x^4+19 x^3+8 x^2+20 x+8$
• $y^2=8 x^6+34 x^5+59 x^4+12 x^3+59 x^2+34 x+8$
• $y^2=39 x^6+21 x^5+55 x^4+16 x^3+39 x^2+13 x+28$
• $y^2=64 x^6+40 x^5+14 x^4+61 x^3+14 x^2+40 x+64$
• $y^2=50 x^6+36 x^5+49 x^4+49 x^3+49 x^2+36 x+50$
• $y^2=54 x^6+23 x^5+54 x^4+62 x^3+5 x^2+44 x+25$
• $y^2=3 x^6+67 x^5+35 x^4+24 x^3+51 x^2+61 x+16$
• $y^2=9 x^6+32 x^5+13 x^4+53 x^3+13 x^2+32 x+9$
• $y^2=8 x^6+2 x^5+51 x^4+27 x^3+51 x^2+2 x+8$
• $y^2=48 x^6+25 x^5+44 x^4+28 x^3+14 x^2+15 x+4$
• $y^2=14 x^6+44 x^4+50 x^3+44 x^2+14$
• $y^2=68 x^6+64 x^5+11 x^4+37 x^3+63 x^2+45 x+33$
• $y^2=49 x^6+65 x^5+63 x^4+26 x^3+39 x^2+46 x+12$
• $y^2=67 x^6+57 x^5+10 x^4+3 x^3+30 x^2+16 x+34$
• $y^2=12 x^6+69 x^5+4 x^4+28 x^3+54 x^2+26 x+24$
• $y^2=49 x^6+59 x^5+45 x^4+41 x^3+24 x^2+7 x+10$
• $y^2=2 x^6+48 x^5+22 x^4+29 x^3+69 x^2+18 x+8$
• $y^2=15 x^6+55 x^5+67 x^4+34 x^2+51 x+27$
• $y^2=9 x^6+53 x^5+42 x^4+35 x^3+39 x^2+62 x+27$
• $y^2=57 x^6+18 x^5+65 x^4+68 x^3+65 x^2+18 x+57$
• $y^2=19 x^6+39 x^5+70 x^4+51 x^3+70 x^2+39 x+19$
• $y^2=56 x^6+27 x^5+53 x^4+3 x^3+53 x^2+27 x+56$
• $y^2=58 x^6+7 x^5+33 x^4+39 x^3+33 x^2+7 x+58$
• $y^2=13 x^6+25 x^5+9 x^4+9 x^3+9 x^2+25 x+13$
• $y^2=59 x^6+4 x^5+x^4+69 x^3+x^2+4 x+59$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{71}$

The isogeny class factors as 1.71.g $\times$ 1.71.i 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.g : \(\Q(\sqrt{-62}) \).
• 1.71.i : \(\Q(\sqrt{-55}) \).

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.ao_hi	$2$	(not in LMFDB)
2.71.ac_dq	$2$	(not in LMFDB)
2.71.c_dq	$2$	(not in LMFDB)