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

• Label: 2.97.ay_ms
• 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:	$1 - 24 x + 330 x^{2} - 2328 x^{3} + 9409 x^{4}$
Frobenius angles:	$\pm0.228703056075$, $\pm0.345832776901$
Angle rank:	$2$ (numerical)
Number field:	\(\Q(\sqrt{-59 +12 \sqrt{2}})\)
Galois group:	$D_{4}$
Jacobians:	$88$
Isomorphism classes:	112
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$7388$	$89335696$	$835670044892$	$7839637564711936$	$73742795819055782108$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$74$	$9494$	$915626$	$88554174$	$8587384874$	$832970943254$	$80798274447818$	$7837433573660158$	$760231058575985930$	$73742412681082248854$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 88 curves (of which all are hyperelliptic):

• $y^2=53 x^6+40 x^5+62 x^4+53 x^3+59 x^2+25 x+63$
• $y^2=21 x^6+53 x^5+90 x^4+89 x^3+81 x^2+35 x+76$
• $y^2=2 x^6+11 x^5+63 x^4+31 x^3+61 x^2+2 x+15$
• $y^2=15 x^6+15 x^5+96 x^4+84 x^3+19 x^2+72 x+56$
• $y^2=19 x^6+58 x^5+23 x^4+59 x^3+x^2+70 x$
• $y^2=57 x^6+56 x^5+40 x^4+37 x^3+84 x^2+x+91$
• $y^2=28 x^6+37 x^5+33 x^4+20 x^3+12 x^2+84 x+21$
• $y^2=87 x^6+33 x^5+x^4+8 x^3+45 x^2+60 x+38$
• $y^2=10 x^6+10 x^5+46 x^4+47 x^3+41 x^2+79 x+77$
• $y^2=6 x^6+10 x^5+80 x^4+4 x^3+53 x^2+42 x+51$
• $y^2=42 x^6+14 x^5+68 x^4+65 x^3+19 x^2+26 x+92$
• $y^2=25 x^6+20 x^5+88 x^4+96 x^3+51 x^2+3 x+12$
• $y^2=40 x^5+44 x^4+94 x^3+28 x^2+68 x+58$
• $y^2=53 x^6+32 x^5+32 x^4+69 x^3+52 x^2+38 x+49$
• $y^2=18 x^6+x^5+8 x^4+92 x^3+30 x^2+84 x+68$
• $y^2=10 x^6+75 x^5+95 x^4+29 x^3+66 x^2+90 x+92$
• $y^2=15 x^6+28 x^5+80 x^4+21 x^3+57 x^2+70$
• $y^2=36 x^6+74 x^5+75 x^4+37 x^3+23 x^2+87 x+28$
• $y^2=42 x^6+86 x^5+50 x^4+74 x^3+13 x^2+32 x+33$
• $y^2=18 x^6+9 x^5+41 x^4+41 x^3+60 x^2+18 x+6$
• and

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

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 \(\Q(\sqrt{-59 +12 \sqrt{2}})\).

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.y_ms	$2$	(not in LMFDB)