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

• Label: 2.97.ay_na
• 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:	$( 1 - 12 x + 97 x^{2} )^{2}$
	$1 - 24 x + 338 x^{2} - 2328 x^{3} + 9409 x^{4}$
Frobenius angles:	$\pm0.291487575149$, $\pm0.291487575149$
Angle rank:	$1$ (numerical)
Jacobians:	$69$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$7396$	$89491600$	$836196855844$	$7840323279360000$	$73742837059284181156$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$74$	$9510$	$916202$	$88561918$	$8587389674$	$832969432230$	$80798248812362$	$7837433415939838$	$760231059972897674$	$73742412722621216550$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 69 curves (of which all are hyperelliptic):

• $y^2=10 x^6+x^5+70 x^4+45 x^3+70 x^2+x+10$
• $y^2=66 x^6+32 x^5+7 x^4+16 x^3+14 x^2+29 x+20$
• $y^2=49 x^6+56 x^5+30 x^4+55 x^3+30 x^2+10 x+60$
• $y^2=15 x^6+49 x^5+42 x^4+38 x^3+17 x^2+11 x+90$
• $y^2=29 x^6+50 x^5+14 x^4+89 x^3+90 x^2+55 x+46$
• $y^2=52 x^6+53 x^4+82 x^3+44 x^2+45$
• $y^2=85 x^6+72 x^5+63 x^4+94 x^3+x^2+86$
• $y^2=75 x^6+45 x^5+49 x^4+59 x^3+44 x^2+56 x+33$
• $y^2=76 x^6+67 x^5+83 x^4+73 x^3+18 x^2+20 x+51$
• $y^2=16 x^6+54 x^5+48 x^4+48 x^3+86 x^2+76 x+55$
• $y^2=86 x^6+61 x^5+4 x^4+62 x^3+44 x^2+17 x+2$
• $y^2=60 x^6+24 x^5+48 x^4+19 x^3+48 x^2+24 x+60$
• $y^2=61 x^6+60 x^5+33 x^4+79 x^3+66 x^2+46 x+3$
• $y^2=95 x^6+2 x^5+26 x^4+45 x^3+x^2+71 x+15$
• $y^2=43 x^6+66 x^5+96 x^4+64 x^3+75 x^2+31 x+24$
• $y^2=45 x^6+87 x^5+15 x^4+94 x^3+55 x^2+38 x+52$
• $y^2=80 x^6+25 x^5+92 x^4+69 x^3+49 x^2+87 x+95$
• $y^2=4 x^6+60 x^5+25 x^4+80 x^3+25 x^2+60 x+4$
• $y^2=56 x^6+35 x^5+89 x^4+68 x^3+72 x^2+79 x+75$
• $y^2=37 x^6+2 x^5+38 x^4+80 x^3+38 x^2+2 x+37$
• and

• $y^2=33 x^6+95 x^5+35 x^4+72 x^3+13 x^2+30 x+23$
• $y^2=12 x^6+74 x^5+71 x^4+34 x^3+71 x^2+74 x+12$
• $y^2=92 x^6+17 x^5+60 x^4+56 x^3+30 x^2+37 x+77$
• $y^2=37 x^6+95 x^5+23 x^4+41 x^3+57 x^2+86 x+13$
• $y^2=34 x^6+34 x^5+82 x^4+41 x^3+12 x^2+51 x+49$
• $y^2=30 x^6+47 x^5+89 x^4+89 x^3+89 x^2+47 x+30$
• $y^2=50 x^6+4 x^5+82 x^4+7 x^3+82 x^2+4 x+50$
• $y^2=83 x^6+95 x^5+96 x^4+47 x^3+9 x^2+24 x+90$
• $y^2=60 x^6+46 x^5+51 x^4+87 x^3+51 x^2+46 x+60$
• $y^2=77 x^6+62 x^5+20 x^4+32 x^3+45 x^2+75 x+68$
• $y^2=66 x^6+69 x^5+18 x^4+23 x^3+81 x^2+15 x+73$
• $y^2=21 x^6+9 x^5+20 x^4+44 x^3+20 x^2+9 x+21$
• $y^2=87 x^6+79 x^5+94 x^4+25 x^3+24 x^2+74 x+16$
• $y^2=45 x^6+6 x^5+39 x^4+96 x^3+30 x^2+84 x+88$
• $y^2=42 x^6+80 x^5+86 x^4+8 x^3+16 x^2+45 x+45$
• $y^2=46 x^6+75 x^5+15 x^4+89 x^3+75 x^2+28 x+43$
• $y^2=4 x^6+19 x^5+12 x^4+45 x^3+12 x^2+19 x+4$
• $y^2=88 x^6+33 x^5+30 x^4+94 x^3+39 x^2+34 x+59$
• $y^2=23 x^6+22 x^5+24 x^4+45 x^3+23 x^2+61 x+87$
• $y^2=37 x^6+21 x^5+32 x^4+x^3+32 x^2+21 x+37$
• $y^2=90 x^6+8 x^5+23 x^4+49 x^3+30 x^2+95 x+37$
• $y^2=89 x^6+37 x^5+88 x^4+70 x^3+88 x^2+37 x+89$
• $y^2=92 x^6+4 x^5+89 x^4+40 x^3+76 x^2+x+43$
• $y^2=19 x^6+37 x^5+51 x^4+78 x^3+41 x^2+53 x+92$
• $y^2=85 x^6+3 x^4+3 x^2+85$
• $y^2=27 x^6+67 x^5+96 x^4+60 x^3+89 x^2+62 x+84$
• $y^2=30 x^6+84 x^5+31 x^4+x^3+11 x^2+67 x+78$
• $y^2=37 x^6+80 x^5+48 x^4+57 x^3+41 x^2+48 x+20$
• $y^2=38 x^6+24 x^5+15 x^4+54 x^3+7 x^2+88 x+37$
• $y^2=90 x^6+44 x^5+37 x^4+11 x^3+93 x^2+23 x+90$
• $y^2=10 x^6+52 x^5+67 x^4+57 x^3+7 x^2+80 x+76$
• $y^2=60 x^6+73 x^5+51 x^4+38 x^3+33 x^2+96 x$
• $y^2=60 x^6+77 x^5+22 x^4+62 x^3+22 x^2+77 x+60$
• $y^2=54 x^6+86 x^5+34 x^4+2 x^3+34 x^2+86 x+54$
• $y^2=52 x^6+71 x^5+25 x^4+42 x^3+25 x^2+71 x+52$
• $y^2=74 x^6+38 x^5+86 x^4+66 x^3+15 x^2+33 x+18$
• $y^2=x^6+7 x^5+13 x^4+77 x^3+13 x^2+7 x+1$
• $y^2=42 x^6+14 x^5+32 x^4+51 x^3+32 x^2+14 x+42$
• $y^2=83 x^6+83 x^5+86 x^4+14 x^3+48 x^2+81 x+69$
• $y^2=95 x^6+50 x^5+23 x^4+19 x^3+23 x^2+50 x+95$
• $y^2=87 x^6+41 x^5+12 x^4+79 x^3+71 x^2+29 x+37$
• $y^2=52 x^6+60 x^5+14 x^4+58 x^3+56 x^2+87 x+30$
• $y^2=24 x^6+18 x^5+74 x^4+71 x^3+21 x^2+45 x+26$
• $y^2=56 x^6+75 x^5+33 x^4+2 x^3+44 x^2+33 x$
• $y^2=63 x^6+51 x^5+47 x^4+12 x^3+77 x^2+77 x+21$
• $y^2=55 x^6+56 x^4+56 x^2+55$
• $y^2=35 x^6+59 x^4+59 x^2+35$
• $y^2=20 x^6+89 x^5+63 x^4+75 x^3+9 x^2+94 x+23$
• $y^2=59 x^6+86 x^5+47 x^4+19 x^3+16 x^2+65 x+51$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{97}$

The isogeny class factors as 1.97.am^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-61}) \)$)$

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_by	$2$	(not in LMFDB)
2.97.y_na	$2$	(not in LMFDB)
2.97.m_bv	$3$	(not in LMFDB)