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

• Label: 2.97.abc_pa
• 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:	$( 1 - 14 x + 97 x^{2} )^{2}$
	$1 - 28 x + 390 x^{2} - 2716 x^{3} + 9409 x^{4}$
Frobenius angles:	$\pm0.248359198326$, $\pm0.248359198326$
Angle rank:	$1$ (numerical)
Jacobians:	$62$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$7056$	$88510464$	$835403312016$	$7840765305225216$	$73744720761342926736$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$70$	$9406$	$915334$	$88566910$	$8587609030$	$832972117822$	$80798259987718$	$7837433240560894$	$760231056076708678$	$73742412687722993086$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 62 curves (of which all are hyperelliptic):

• $y^2=64 x^6+43 x^5+93 x^4+78 x^3+93 x^2+43 x+64$
• $y^2=91 x^6+91 x^5+73 x^4+4 x^3+66 x^2+88 x+49$
• $y^2=6 x^6+82 x^5+65 x^4+40 x^3+79 x^2+7 x+93$
• $y^2=x^6+36 x^5+44 x^4+35 x^3+44 x^2+36 x+1$
• $y^2=22 x^6+78 x^5+28 x^4+68 x^3+74 x^2+41 x+8$
• $y^2=81 x^6+31 x^5+79 x^4+11 x^3+79 x^2+31 x+81$
• $y^2=59 x^6+37 x^5+40 x^4+62 x^3+83 x^2+46 x+5$
• $y^2=43 x^6+33 x^5+63 x^4+51 x^3+63 x^2+33 x+43$
• $y^2=73 x^6+50 x^5+34 x^4+25 x^3+82 x^2+68 x+37$
• $y^2=41 x^6+94 x^4+94 x^2+41$
• $y^2=59 x^6+88 x^5+92 x^4+69 x^3+92 x^2+88 x+59$
• $y^2=50 x^6+3 x^5+22 x^4+16 x^3+22 x^2+3 x+50$
• $y^2=62 x^6+86 x^4+86 x^2+62$
• $y^2=87 x^6+44 x^5+43 x^4+46 x^3+43 x^2+44 x+87$
• $y^2=5 x^6+4 x^5+36 x^4+79 x^3+36 x^2+4 x+5$
• $y^2=20 x^6+25 x^5+49 x^4+90 x^3+49 x^2+25 x+20$
• $y^2=15 x^6+62 x^4+62 x^2+15$
• $y^2=52 x^6+57 x^5+43 x^4+10 x^3+49 x^2+63 x+63$
• $y^2=85 x^6+88 x^5+8 x^4+44 x^3+8 x^2+88 x+85$
• $y^2=34 x^6+61 x^5+78 x^4+52 x^3+78 x^2+61 x+34$
• and

• $y^2=91 x^6+45 x^5+x^4+3 x^3+81 x^2+74 x+35$
• $y^2=48 x^6+58 x^5+9 x^4+92 x^3+64 x^2+90 x+48$
• $y^2=15 x^6+50 x^5+8 x^4+63 x^3+11 x^2+43 x+82$
• $y^2=26 x^6+26 x^5+41 x^4+30 x^3+41 x^2+26 x+26$
• $y^2=33 x^6+22 x^5+46 x^4+96 x^3+2 x^2+30 x+5$
• $y^2=18 x^6+44 x^5+58 x^4+50 x^3+85 x^2+81 x+5$
• $y^2=50 x^6+81 x^4+81 x^2+50$
• $y^2=47 x^6+81 x^5+18 x^4+65 x^3+35 x^2+67 x+1$
• $y^2=x^6+44 x^3+18$
• $y^2=58 x^6+69 x^5+25 x^4+44 x^3+25 x^2+69 x+58$
• $y^2=82 x^6+30 x^5+29 x^4+5 x^3+29 x^2+30 x+82$
• $y^2=82 x^6+30 x^5+78 x^4+83 x^3+79 x^2+50 x+45$
• $y^2=37 x^6+56 x^5+34 x^3+5 x^2+45 x+36$
• $y^2=74 x^6+45 x^5+55 x^4+6 x^3+55 x^2+45 x+74$
• $y^2=90 x^6+39 x^5+71 x^4+17 x^3+35 x^2+85 x+20$
• $y^2=68 x^6+81 x^5+9 x^4+59 x^3+65 x^2+54 x+77$
• $y^2=76 x^6+48 x^4+48 x^2+76$
• $y^2=20 x^6+71 x^5+36 x^4+80 x^3+65 x^2+37 x+21$
• $y^2=21 x^6+68 x^5+58 x^4+53 x^3+4 x^2+4 x+34$
• $y^2=x^6+x^3+27$
• $y^2=82 x^6+59 x^5+78 x^4+64 x^3+79 x^2+48 x+55$
• $y^2=16 x^6+25 x^4+25 x^2+16$
• $y^2=50 x^6+30 x^5+61 x^4+54 x^3+61 x^2+30 x+50$
• $y^2=73 x^6+13 x^5+83 x^4+81 x^3+21 x^2+5 x+81$
• $y^2=62 x^6+78 x^5+51 x^4+19 x^3+84 x^2+76 x+48$
• $y^2=91 x^6+85 x^5+27 x^4+46 x^3+27 x^2+85 x+91$
• $y^2=78 x^6+36 x^5+18 x^4+47 x^3+16 x^2+50 x+46$
• $y^2=52 x^6+20 x^5+39 x^4+46 x^3+39 x^2+20 x+52$
• $y^2=62 x^6+57 x^5+34 x^4+87 x^3+77 x^2+60 x+49$
• $y^2=x^6+96$
• $y^2=8 x^6+27 x^5+24 x^4+58 x^3+74 x^2+24 x+74$
• $y^2=78 x^6+62 x^5+46 x^4+20 x^3+46 x^2+62 x+78$
• $y^2=73 x^6+29 x^5+34 x^4+6 x^3+34 x^2+29 x+73$
• $y^2=x^6+9 x^3+50$
• $y^2=65 x^6+64 x^5+12 x^4+79 x^3+40 x^2+47 x+34$
• $y^2=6 x^6+x^5+5 x^4+77 x^3+91 x^2+53 x+16$
• $y^2=23 x^6+58 x^5+51 x^4+85 x^3+85 x+2$
• $y^2=84 x^6+74 x^5+69 x^4+12 x^3+69 x^2+74 x+84$
• $y^2=23 x^6+12 x^5+59 x^4+19 x^3+84 x^2+30 x+20$
• $y^2=20 x^6+72 x^5+56 x^4+91 x^3+76 x^2+95 x+77$
• $y^2=67 x^6+27 x^5+58 x^4+90 x^3+28 x^2+65 x+77$
• $y^2=48 x^6+48 x^5+23 x^4+90 x^3+23 x^2+48 x+48$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{97}$

The isogeny class factors as 1.97.ao^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

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_ac	$2$	(not in LMFDB)
2.97.bc_pa	$2$	(not in LMFDB)
2.97.at_ke	$3$	(not in LMFDB)
2.97.ak_il	$3$	(not in LMFDB)
2.97.f_acu	$3$	(not in LMFDB)
2.97.o_dv	$3$	(not in LMFDB)
2.97.bm_vj	$3$	(not in LMFDB)