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

• Label: 2.97.a_gw
• 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:	$( 1 - 4 x + 97 x^{2} )( 1 + 4 x + 97 x^{2} )$
	$1 + 178 x^{2} + 9409 x^{4}$
Frobenius angles:	$\pm0.434908349536$, $\pm0.565091650464$
Angle rank:	$1$ (numerical)
Jacobians:	$112$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$9588$	$91929744$	$832972620276$	$7835155901485056$	$73742412681651655668$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$98$	$9766$	$912674$	$88503550$	$8587340258$	$832973235622$	$80798284478114$	$7837433617426174$	$760231058654565218$	$73742412673810485286$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 112 curves (of which all are hyperelliptic):

• $y^2=90 x^6+14 x^5+30 x^4+88 x^3+92 x+59$
• $y^2=62 x^6+70 x^5+53 x^4+52 x^3+72 x+4$
• $y^2=56 x^6+65 x^5+25 x^4+56 x^3+28 x^2+23 x+77$
• $y^2=86 x^6+34 x^5+28 x^4+86 x^3+43 x^2+18 x+94$
• $y^2=60 x^6+83 x^5+60 x^4+70 x^3+14 x^2+44 x+1$
• $y^2=9 x^6+27 x^5+9 x^4+59 x^3+70 x^2+26 x+5$
• $y^2=19 x^6+92 x^5+35 x^4+44 x^3+80 x^2+27 x+83$
• $y^2=91 x^5+69 x^4+73 x^3+13 x^2+43 x$
• $y^2=67 x^5+54 x^4+74 x^3+65 x^2+21 x$
• $y^2=75 x^6+13 x^5+79 x^4+86 x^3+45 x^2+29 x+79$
• $y^2=71 x^6+33 x^5+28 x^4+17 x^3+53 x^2+32 x+35$
• $y^2=88 x^6+42 x^5+19 x^4+71 x^3+78 x^2+6 x+64$
• $y^2=52 x^6+16 x^5+95 x^4+64 x^3+2 x^2+30 x+29$
• $y^2=x^6+28 x^5+93 x^4+30 x^3+22 x^2+68 x+13$
• $y^2=5 x^6+43 x^5+77 x^4+53 x^3+13 x^2+49 x+65$
• $y^2=52 x^6+86 x^5+42 x^4+3 x^3+42 x^2+86 x+52$
• $y^2=66 x^6+42 x^5+16 x^4+15 x^3+16 x^2+42 x+66$
• $y^2=31 x^6+87 x^5+87 x^4+92 x^3+31 x^2+87 x+59$
• $y^2=58 x^6+47 x^5+47 x^4+72 x^3+58 x^2+47 x+4$
• $y^2=72 x^6+44 x^5+66 x^4+59 x^3+71 x^2+47 x+17$
• and

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{97^{2}}$.

Endomorphism algebra over $\F_{97}$

The isogeny class factors as 1.97.ae $\times$ 1.97.e and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.97.ae : \(\Q(\sqrt{-93}) \).
• 1.97.e : \(\Q(\sqrt{-93}) \).

Endomorphism algebra over $\overline{\F}_{97}$

The base change of $A$ to $\F_{97^{2}}$ is 1.9409.gw^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-93}) \)$)$

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.ai_ic	$2$	(not in LMFDB)
2.97.i_ic	$2$	(not in LMFDB)
2.97.a_agw	$4$	(not in LMFDB)