Abelian variety isogeny class 2.97.au_li over $\F_{97}$

• Label: 2.97.au_li
• Base field: $\F_{97}$
• 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:	$\F_{97}$
Dimension:	$2$
L-polynomial:	$( 1 - 10 x + 97 x^{2} )^{2}$
	$1 - 20 x + 294 x^{2} - 1940 x^{3} + 9409 x^{4}$
Frobenius angles:	$\pm0.330505784077$, $\pm0.330505784077$
Angle rank:	$1$ (numerical)
Jacobians:	$112$
Cyclic group of points:	no
Non-cyclic primes:	$2, 11$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$7744$	$90326016$	$836463893056$	$7839201269661696$	$73740945137519116864$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$78$	$9598$	$916494$	$88549246$	$8587169358$	$832968359422$	$80798264600334$	$7837433749213438$	$760231062131074638$	$73742412709238782078$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 112 curves (of which all are hyperelliptic):

• $y^2=59 x^6+39 x^5+89 x^3+28 x+84$
• $y^2=89 x^6+36 x^5+78 x^4+53 x^3+94 x^2+17 x+52$
• $y^2=29 x^6+74 x^5+89 x^4+52 x^3+46 x^2+86 x+40$
• $y^2=24 x^6+30 x^5+68 x^4+43 x^3+2 x^2+46 x+78$
• $y^2=60 x^6+63 x^5+8 x^4+55 x^3+8 x^2+63 x+60$
• $y^2=77 x^6+63 x^5+45 x^4+23 x^3+45 x^2+63 x+77$
• $y^2=77 x^6+26 x^5+52 x^4+2 x^3+48 x^2+35 x+66$
• $y^2=35 x^6+67 x^5+42 x^4+47 x^3+63 x^2+52 x+78$
• $y^2=49 x^6+44 x^5+x^4+12 x^3+76 x^2+71 x+75$
• $y^2=96 x^6+12 x^5+39 x^4+33 x^3+34 x^2+56 x+49$
• $y^2=52 x^6+58 x^5+64 x^4+39 x^3+64 x^2+58 x+52$
• $y^2=12 x^6+69 x^5+44 x^4+x^2+7 x+33$
• $y^2=82 x^5+7 x^3+28 x^2+38 x+6$
• $y^2=22 x^6+20 x^5+31 x^4+83 x^3+42 x^2+x+24$
• $y^2=65 x^6+91 x^5+85 x^4+70 x^3+36 x^2+43 x+88$
• $y^2=24 x^6+63 x^5+58 x^4+54 x^3+13 x^2+7 x+53$
• $y^2=5 x^6+58 x^3+60$
• $y^2=94 x^6+65 x^5+58 x^4+40 x^3+58 x^2+65 x+94$
• $y^2=45 x^6+96 x^5+76 x^4+81 x^3+22 x^2+82 x+61$
• $y^2=79 x^6+16 x^5+60 x^4+51 x^3+47 x^2+57 x+37$
• and

• $y^2=63 x^6+36 x^5+77 x^4+51 x^3+85 x^2+15 x+61$
• $y^2=48 x^6+5 x^5+43 x^4+79 x^3+43 x^2+5 x+48$
• $y^2=30 x^6+61 x^5+61 x^4+70 x^3+87 x^2+74 x+18$
• $y^2=68 x^6+56 x^4+56 x^2+68$
• $y^2=25 x^6+51 x^5+67 x^4+37 x^3+26 x^2+59 x+6$
• $y^2=10 x^6+38 x^5+46 x^4+95 x^3+79 x^2+55 x+49$
• $y^2=19 x^6+8 x^5+3 x^4+13 x^3+3 x^2+8 x+19$
• $y^2=59 x^6+54 x^5+96 x^4+49 x^3+53 x^2+75 x+92$
• $y^2=14 x^6+61 x^5+28 x^4+77 x^3+76 x^2+4 x+82$
• $y^2=94 x^6+75 x^5+67 x^4+91 x^3+92 x^2+91 x+66$
• $y^2=57 x^6+7 x^5+3 x^4+89 x^3+76 x^2+2 x+86$
• $y^2=13 x^6+84 x^5+12 x^4+67 x^3+57 x^2+30 x+66$
• $y^2=36 x^6+59 x^5+13 x^4+82 x^3+25 x^2+96 x+20$
• $y^2=18 x^6+64 x^5+7 x^4+61 x^3+35 x^2+52 x+84$
• $y^2=56 x^6+40 x^5+79 x^4+58 x^3+79 x^2+40 x+56$
• $y^2=25 x^6+39 x^5+52 x^4+32 x^3+77 x^2+11 x+8$
• $y^2=29 x^6+6 x^5+31 x^4+65 x^3+4 x^2+75 x+16$
• $y^2=70 x^6+46 x^5+21 x^4+96 x^3+23 x^2+15 x+73$
• $y^2=28 x^6+57 x^5+39 x^4+93 x^3+22 x^2+13 x+87$
• $y^2=8 x^6+61 x^5+23 x^4+30 x^3+21 x^2+36 x+18$
• $y^2=45 x^6+36 x^5+83 x^4+17 x^3+8 x^2+22 x+57$
• $y^2=5 x^6+5 x^3+68$
• $y^2=63 x^6+85 x^5+21 x^4+17 x^3+21 x^2+85 x+63$
• $y^2=22 x^6+64 x^5+79 x^4+13 x^3+40 x^2+69 x+67$
• $y^2=16 x^6+31 x^5+32 x^4+53 x^3+22 x^2+12 x+95$
• $y^2=37 x^6+68 x^5+12 x^4+92 x^3+94 x^2+44 x+21$
• $y^2=46 x^6+20 x^5+80 x^4+47 x^3+80 x^2+20 x+46$
• $y^2=56 x^6+23 x^5+26 x^4+57 x^3+47 x^2+2 x+13$
• $y^2=57 x^6+14 x^4+14 x^2+57$
• $y^2=68 x^6+86 x^5+84 x^4+35 x^3+38 x^2+65 x+37$
• $y^2=x^5+96 x$
• $y^2=86 x^6+27 x^5+60 x^4+71 x^3+74 x^2+11 x+88$
• $y^2=17 x^6+44 x^5+28 x^4+33 x^3+24 x^2+6 x+41$
• $y^2=69 x^6+65 x^5+83 x^4+89 x^3+28 x^2+66 x+30$
• $y^2=65 x^6+71 x^4+71 x^2+65$
• $y^2=9 x^6+63 x^5+52 x^4+32 x^3+69 x^2+55 x+48$
• $y^2=5 x^6+52 x^3+41$
• $y^2=73 x^5+48 x^4+47 x^3+50 x^2+75 x$
• $y^2=80 x^6+38 x^5+31 x^4+57 x^3+81 x^2+79 x+78$
• $y^2=74 x^6+86 x^5+22 x^4+20 x^3+54 x^2+66 x+83$
• $y^2=68 x^6+3 x^5+21 x^4+80 x^3+89 x^2+41 x+28$
• $y^2=94 x^6+23 x^4+23 x^2+94$
• $y^2=20 x^6+36 x^5+87 x^4+72 x^3+2 x^2+63 x+30$
• $y^2=36 x^6+54 x^4+54 x^2+36$
• $y^2=15 x^6+54 x^5+27 x^4+5 x^3+18 x^2+8 x+59$
• $y^2=85 x^6+44 x^5+21 x^4+75 x^3+51 x^2+18 x+75$
• $y^2=5 x^6+67 x^3+84$
• $y^2=38 x^6+61 x^5+58 x^4+5 x^3+58 x^2+61 x+38$
• $y^2=87 x^6+26 x^5+92 x^4+34 x^3+11 x^2+24 x+4$
• $y^2=94 x^6+49 x^5+16 x^3+86 x+95$
• $y^2=78 x^6+68 x^5+38 x^4+34 x^3+38 x^2+4 x+55$
• $y^2=37 x^6+64 x^5+94 x^4+15 x^3+94 x^2+64 x+37$
• $y^2=24 x^6+95 x^5+48 x^4+76 x^3+68 x^2+93 x+66$
• $y^2=5 x^6+5 x^3+41$
• $y^2=52 x^6+4 x^5+17 x^4+48 x^3+39 x^2+62 x+46$
• $y^2=80 x^6+66 x^5+86 x^4+27 x^3+2 x^2+45 x+69$
• $y^2=55 x^6+70 x^5+29 x^4+42 x^3+70 x^2+25 x+11$
• $y^2=37 x^6+71 x^5+41 x^4+79 x^3+27 x^2+24 x+20$
• $y^2=76 x^6+56 x^5+39 x^4+10 x^3+67 x^2+81 x+82$
• $y^2=88 x^6+11 x^5+30 x^4+52 x^3+30 x^2+11 x+88$
• $y^2=5 x^6+5 x^3+13$
• $y^2=5 x^6+30 x^3+13$
• $y^2=62 x^6+75 x^5+12 x^4+7 x^3+85 x^2+75 x+35$
• $y^2=20 x^6+44 x^5+54 x^4+47 x^3+23 x^2+20 x+3$
• $y^2=34 x^6+20 x^5+65 x^4+56 x^3+x^2+7 x+72$
• $y^2=10 x^6+26 x^5+60 x^4+6 x^3+21 x^2+50 x+94$
• $y^2=24 x^6+30 x^5+59 x^4+41 x^3+54 x^2+62 x+84$
• $y^2=23 x^6+85 x^5+46 x^4+71 x^3+60 x^2+72 x+14$
• $y^2=29 x^6+74 x^5+69 x^4+48 x^3+83 x^2+67 x+40$
• $y^2=40 x^6+88 x^5+30 x^4+5 x^3+78 x^2+9 x+90$
• $y^2=16 x^6+11 x^5+33 x^4+38 x^3+88 x^2+89 x+16$
• $y^2=38 x^6+71 x^5+92 x^4+59 x^3+79 x^2+79 x+78$
• $y^2=5 x^6+46 x^3+60$
• $y^2=91 x^6+18 x^5+77 x^4+42 x^3+84 x^2+11 x+93$
• $y^2=55 x^6+79 x^5+93 x^4+44 x^3+93 x^2+79 x+55$
• $y^2=57 x^6+42 x^5+2 x^4+95 x^3+35 x^2+10 x+92$
• $y^2=96 x^6+85 x^5+37 x^4+89 x^3+6 x^2+33 x+48$
• $y^2=65 x^6+x^5+49 x^4+7 x^3+8 x^2+29 x+91$
• $y^2=87 x^6+60 x^5+16 x^4+84 x^3+49 x^2+55 x+76$
• $y^2=30 x^6+91 x^5+91 x^3+39 x^2+51 x+86$
• $y^2=84 x^6+96 x^5+86 x^4+35 x^3+10 x^2+38 x+67$
• $y^2=56 x^6+26 x^5+93 x^4+68 x^3+60 x^2+58 x+15$
• $y^2=39 x^6+49 x^5+37 x^4+46 x^3+2 x^2+7 x+46$
• $y^2=71 x^6+87 x^5+7 x^4+57 x^3+92 x^2+88 x+13$
• $y^2=55 x^6+40 x^5+53 x^4+25 x^3+53 x^2+40 x+55$
• $y^2=60 x^6+31 x^4+31 x^2+60$
• $y^2=59 x^6+21 x^5+90 x^4+27 x^3+41 x^2+83 x+41$
• $y^2=48 x^6+6 x^5+17 x^4+34 x^3+75 x^2+70 x+80$
• $y^2=65 x^6+6 x^4+6 x^2+65$
• $y^2=34 x^6+38 x^5+66 x^4+75 x^3+24 x^2+90 x+20$
• $y^2=30 x^6+72 x^5+47 x^4+43 x^3+30 x^2+93 x+88$
• $y^2=68 x^6+65 x^5+58 x^4+69 x^3+10 x^2+85 x+21$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{97}$.

Endomorphism algebra over $\F_{97}$

The isogeny class factors as 1.97.ak^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$

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
2.97.a_dq	$2$	(not in LMFDB)
2.97.u_li	$2$	(not in LMFDB)
2.97.k_d	$3$	(not in LMFDB)