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

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

Invariants

Base field:	$\F_{97}$
Dimension:	$2$
L-polynomial:	$1 - 178 x^{2} + 9409 x^{4}$
Frobenius angles:	$\pm0.0650916504636$, $\pm0.934908349536$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(i, \sqrt{93})\)
Galois group:	$C_2^2$
Jacobians:	$42$
Cyclic group of points:	no
Non-cyclic primes:	$2$

This isogeny class is simple but not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$9232$	$85229824$	$832971389584$	$7835155901485056$	$73742412697333996432$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$98$	$9054$	$912674$	$88503550$	$8587340258$	$832970774238$	$80798284478114$	$7837433617426174$	$760231058654565218$	$73742412705175166814$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 42 curves (of which all are hyperelliptic):

• $y^2=40 x^6+77 x^5+87 x^4+32 x^3+44 x^2+59 x+73$
• $y^2=23 x^6+92 x^5+88 x^4+88 x^3+28 x^2+51 x+25$
• $y^2=84 x^6+81 x^5+75 x^4+87 x^3+46 x^2+69$
• $y^2=51 x^6+87 x^5+42 x^4+67 x^3+49 x^2+61 x+63$
• $y^2=12 x^6+76 x^5+29 x^4+23 x^2+56 x+12$
• $y^2=66 x^6+40 x^5+34 x^4+67 x^3+66 x^2+45 x+40$
• $y^2=82 x^6+61 x^5+16 x^4+92 x^3+70 x^2+47 x+34$
• $y^2=32 x^6+16 x^5+24 x^4+6 x^3+93 x^2+95 x+50$
• $y^2=8 x^6+42 x^5+11 x^4+70 x^3+29 x^2+64 x+39$
• $y^2=5 x^6+16 x^5+23 x^4+32 x^3+46 x^2+30 x+60$
• $y^2=11 x^6+26 x^5+4 x^4+87 x^3+56 x^2+52 x+17$
• $y^2=78 x^6+62 x^5+8 x^4+14 x^3+38 x^2+58 x+87$
• $y^2=8 x^6+30 x^5+33 x^4+60 x^3+13 x^2+10 x+52$
• $y^2=70 x^6+22 x^5+29 x^4+44 x^3+76 x^2+81 x+67$
• $y^2=59 x^6+13 x^5+48 x^4+26 x^3+89 x^2+17 x+44$
• $y^2=72 x^6+77 x^5+66 x^4+45 x^3+14 x^2+26 x+26$
• $y^2=23 x^6+69 x^5+17 x^4+71 x^3+33 x^2+17 x+47$
• $y^2=18 x^6+54 x^5+85 x^4+64 x^3+68 x^2+85 x+41$
• $y^2=91 x^6+35 x^5+3 x^4+42 x^3+81 x^2+42 x+55$
• $y^2=89 x^6+31 x^5+3 x^4+16 x^3+85 x^2+42 x+38$
• and

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{97}$

The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{93})\).

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

The base change of $A$ to $\F_{97^{2}}$ is 1.9409.agw^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	$4$	(not in LMFDB)
2.97.a_gw	$4$	(not in LMFDB)
2.97.i_ic	$4$	(not in LMFDB)