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

• Label: 2.97.u_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:	$\pm0.669494215923$, $\pm0.669494215923$
Angle rank:	$1$ (numerical)
Jacobians:	$112$
Cyclic group of points:	no
Non-cyclic primes:	$2, 3$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$11664$	$90326016$	$829491063696$	$7839201269661696$	$73743880290419301264$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$118$	$9598$	$908854$	$88549246$	$8587511158$	$832968359422$	$80798304355894$	$7837433749213438$	$760231055178055798$	$73742412709238782078$

Jacobians and polarizations

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

• $y^2=70 x^6+66 x^5+76 x^3+25 x+75$
• $y^2=76 x^6+46 x^5+35 x^4+30 x^3+77 x^2+81 x+88$
• $y^2=64 x^6+73 x^5+76 x^4+88 x^3+48 x^2+56 x+8$
• $y^2=23 x^6+53 x^5+49 x^4+21 x^3+10 x^2+36 x+2$
• $y^2=12 x^6+32 x^5+21 x^4+11 x^3+21 x^2+32 x+12$
• $y^2=93 x^6+32 x^5+9 x^4+24 x^3+9 x^2+32 x+93$
• $y^2=94 x^6+33 x^5+66 x^4+10 x^3+46 x^2+78 x+39$
• $y^2=7 x^6+91 x^5+86 x^4+87 x^3+32 x^2+88 x+35$
• $y^2=68 x^6+67 x^5+39 x^4+80 x^3+54 x^2+53 x+15$
• $y^2=92 x^6+60 x^5+x^4+68 x^3+73 x^2+86 x+51$
• $y^2=66 x^6+96 x^5+29 x^4+x^3+29 x^2+96 x+66$
• $y^2=60 x^6+54 x^5+26 x^4+5 x^2+35 x+68$
• $y^2=22 x^5+35 x^3+43 x^2+93 x+30$
• $y^2=13 x^6+3 x^5+58 x^4+27 x^3+16 x^2+5 x+23$
• $y^2=13 x^6+57 x^5+17 x^4+14 x^3+46 x^2+28 x+37$
• $y^2=23 x^6+24 x^5+96 x^4+76 x^3+65 x^2+35 x+71$
• $y^2=x^6+31 x^3+12$
• $y^2=82 x^6+34 x^5+96 x^4+6 x^3+96 x^2+34 x+82$
• $y^2=9 x^6+58 x^5+54 x^4+55 x^3+82 x^2+94 x+51$
• $y^2=74 x^6+42 x^5+12 x^4+49 x^3+87 x^2+89 x+85$
• and

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{97}$

The isogeny class factors as 1.97.k^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.au_li	$2$	(not in LMFDB)
2.97.a_dq	$2$	(not in LMFDB)
2.97.ak_d	$3$	(not in LMFDB)