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

• Label: 2.97.am_iw
• 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:	$( 1 - 6 x + 97 x^{2} )^{2}$
	$1 - 12 x + 230 x^{2} - 1164 x^{3} + 9409 x^{4}$
Frobenius angles:	$\pm0.401475932553$, $\pm0.401475932553$
Angle rank:	$1$ (numerical)
Jacobians:	$92$
Cyclic group of points:	no
Non-cyclic primes:	$2, 23$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$8464$	$91546624$	$835768953616$	$7836345607274496$	$73739230513144320784$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$86$	$9726$	$915734$	$88516990$	$8586969686$	$832970973822$	$80798314236950$	$7837433872947454$	$760231057439381078$	$73742412655180383486$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 92 curves (of which all are hyperelliptic):

• $y^2=76 x^6+52 x^5+62 x^4+43 x^3+95 x^2+11 x+83$
• $y^2=56 x^6+87 x^5+21 x^4+72 x^3+10 x^2+96 x+37$
• $y^2=21 x^6+93 x^5+36 x^4+74 x^3+8 x^2+x+15$
• $y^2=42 x^6+35 x^5+33 x^4+30 x^3+23 x^2+42 x+56$
• $y^2=45 x^6+58 x^5+93 x^4+76 x^3+73 x^2+2 x+43$
• $y^2=74 x^6+10 x^5+80 x^4+8 x^3+82 x^2+39 x+15$
• $y^2=28 x^6+94 x^4+94 x^2+28$
• $y^2=3 x^6+25 x^4+25 x^2+3$
• $y^2=49 x^6+48 x^5+76 x^4+57 x^3+41 x^2+41 x+44$
• $y^2=87 x^6+74 x^5+91 x^4+83 x^3+59 x+14$
• $y^2=33 x^6+17 x^5+36 x^4+85 x^3+35 x^2+82 x+45$
• $y^2=60 x^6+x^5+21 x^4+15 x^3+87 x^2+85 x+90$
• $y^2=45 x^6+63 x^5+62 x^4+40 x^3+4 x^2+39 x+56$
• $y^2=77 x^6+91 x^5+30 x^4+69 x^3+80 x^2+68 x+12$
• $y^2=45 x^6+54 x^5+32 x^4+86 x^3+55 x+71$
• $y^2=26 x^6+45 x^5+33 x^4+11 x^3+5 x^2+55 x+19$
• $y^2=80 x^6+68 x^5+9 x^4+36 x^3+55 x^2+44 x+4$
• $y^2=82 x^6+69 x^5+89 x^4+52 x^3+48 x^2+59 x+39$
• $y^2=5 x^6+90 x^5+34 x^4+96 x^3+41 x^2+8 x+17$
• $y^2=94 x^6+73 x^5+23 x^4+17 x^3+41 x^2+11 x+49$
• and

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{97}$

The isogeny class factors as 1.97.ag^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-22}) \)$)$

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_gc	$2$	(not in LMFDB)
2.97.m_iw	$2$	(not in LMFDB)
2.97.g_acj	$3$	(not in LMFDB)