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

• Label: 2.97.d_aeq
• Base field: $\F_{97}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{97}$
Dimension:	$2$
L-polynomial:	$1 + 3 x - 120 x^{2} + 291 x^{3} + 9409 x^{4}$
Frobenius angles:	$\pm0.190235007350$, $\pm0.934600127152$
Angle rank:	$2$ (numerical)
Number field:	\(\Q(\sqrt{-278 +6 \sqrt{1265}})\)
Galois group:	$D_{4}$
Jacobians:	$84$
Cyclic group of points:	no
Non-cyclic primes:	$2$

This isogeny class is simple and geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$9584$	$86217664$	$834781581056$	$7837517078300416$	$73744808617916131184$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$101$	$9161$	$914654$	$88530225$	$8587619261$	$832973047742$	$80798295925805$	$7837433595108769$	$760231057097495198$	$73742412681108756761$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 84 curves (of which all are hyperelliptic):

• $y^2=66 x^6+75 x^5+83 x^4+62 x^3+5 x^2+40 x+36$
• $y^2=28 x^6+37 x^5+63 x^4+12 x^3+85 x^2+24 x+11$
• $y^2=22 x^6+93 x^5+59 x^4+89 x^3+70 x^2+81 x+36$
• $y^2=19 x^6+4 x^5+59 x^4+3 x^3+86 x^2+77 x+32$
• $y^2=39 x^6+68 x^5+37 x^4+40 x^3+55 x^2+76 x+73$
• $y^2=90 x^5+85 x^4+33 x^3+24 x+93$
• $y^2=26 x^5+14 x^4+77 x^3+62 x^2+42 x+25$
• $y^2=4 x^6+3 x^5+64 x^4+57 x^3+28 x^2+32 x+32$
• $y^2=4 x^6+x^5+15 x^4+5 x^3+6 x^2+90 x$
• $y^2=65 x^6+43 x^5+58 x^4+31 x^3+80 x^2+46 x+71$
• $y^2=19 x^6+10 x^5+46 x^4+81 x^3+45 x^2+62 x+13$
• $y^2=84 x^6+96 x^5+82 x^4+18 x^3+22 x^2+91 x+4$
• $y^2=11 x^6+32 x^5+46 x^4+80 x^3+64 x^2+75 x+52$
• $y^2=90 x^6+x^5+8 x^4+93 x^3+50 x^2+15 x+34$
• $y^2=11 x^6+78 x^5+46 x^4+x^3+12 x^2+31 x+72$
• $y^2=4 x^6+41 x^5+89 x^4+84 x^3+20 x^2+86 x+89$
• $y^2=92 x^6+92 x^5+50 x^4+66 x^3+6 x^2+94 x+41$
• $y^2=6 x^6+33 x^5+60 x^4+30 x^3+90 x^2+84 x+15$
• $y^2=69 x^6+26 x^4+41 x^3+57 x^2+86 x+24$
• $y^2=41 x^6+45 x^5+92 x^4+3 x^3+11 x^2+40 x+54$
• and

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{97}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-278 +6 \sqrt{1265}})\).

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
2.97.ad_aeq	$2$	(not in LMFDB)