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

• Label: 2.97.y_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:	$\pm0.708512424851$, $\pm0.708512424851$
Angle rank:	$1$ (numerical)
Jacobians:	$69$
Cyclic group of points:	no
Non-cyclic primes:	$2, 5, 11$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$12100$	$89491600$	$829757028100$	$7840323279360000$	$73741988355271802500$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$122$	$9510$	$909146$	$88561918$	$8587290842$	$832969432230$	$80798320143866$	$7837433415939838$	$760231057336232762$	$73742412722621216550$

Jacobians and polarizations

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

• $y^2=2 x^6+39 x^5+14 x^4+9 x^3+14 x^2+39 x+2$
• $y^2=39 x^6+63 x^5+35 x^4+80 x^3+70 x^2+48 x+3$
• $y^2=51 x^6+86 x^5+53 x^4+81 x^3+53 x^2+50 x+9$
• $y^2=3 x^6+68 x^5+86 x^4+27 x^3+81 x^2+41 x+18$
• $y^2=48 x^6+56 x^5+70 x^4+57 x^3+62 x^2+81 x+36$
• $y^2=88 x^6+30 x^4+94 x^3+67 x^2+9$
• $y^2=17 x^6+92 x^5+32 x^4+77 x^3+39 x^2+56$
• $y^2=15 x^6+9 x^5+68 x^4+70 x^3+67 x^2+50 x+26$
• $y^2=89 x^6+44 x^5+27 x^4+74 x^3+90 x^2+3 x+61$
• $y^2=42 x^6+69 x^5+29 x^4+29 x^3+56 x^2+54 x+11$
• $y^2=42 x^6+14 x^5+20 x^4+19 x^3+26 x^2+85 x+10$
• $y^2=9 x^6+23 x^5+46 x^4+95 x^3+46 x^2+23 x+9$
• $y^2=51 x^6+12 x^5+26 x^4+74 x^3+52 x^2+48 x+20$
• $y^2=19 x^6+78 x^5+44 x^4+9 x^3+39 x^2+53 x+3$
• $y^2=28 x^6+52 x^5+58 x^4+71 x^3+15 x^2+45 x+63$
• $y^2=31 x^6+47 x^5+75 x^4+82 x^3+81 x^2+93 x+66$
• $y^2=12 x^6+28 x^5+72 x^4+54 x^3+51 x^2+47 x+87$
• $y^2=20 x^6+9 x^5+28 x^4+12 x^3+28 x^2+9 x+20$
• $y^2=50 x^6+7 x^5+76 x^4+33 x^3+92 x^2+74 x+15$
• $y^2=85 x^6+78 x^5+27 x^4+16 x^3+27 x^2+78 x+85$
• and

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{97}$

The isogeny class factors as 1.97.m^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.ay_na	$2$	(not in LMFDB)
2.97.a_by	$2$	(not in LMFDB)
2.97.am_bv	$3$	(not in LMFDB)