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

• Label: 2.97.a_agu
• 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:	$1 - 176 x^{2} + 9409 x^{4}$
Frobenius angles:	$\pm0.0691014388248$, $\pm0.930898561175$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-2}, \sqrt{-185})\)
Galois group:	$C_2^2$
Jacobians:	$48$
Isomorphism classes:	288
Cyclic group of points:	no
Non-cyclic primes:	$3$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$9234$	$85266756$	$832971521106$	$7835281241231376$	$73742412699191647314$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$98$	$9058$	$912674$	$88504966$	$8587340258$	$832971037282$	$80798284478114$	$7837433652860158$	$760231058654565218$	$73742412708890468578$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 48 curves (of which all are hyperelliptic):

• $y^2=82 x^6+73 x^5+78 x^4+81 x^3+8 x^2+90 x+16$
• $y^2=22 x^6+74 x^5+2 x^4+17 x^3+40 x^2+62 x+80$
• $y^2=78 x^6+25 x^5+70 x^4+27 x^3+26 x^2+96 x+44$
• $y^2=2 x^6+28 x^5+59 x^4+38 x^3+33 x^2+92 x+26$
• $y^2=95 x^6+54 x^5+55 x^4+72 x^3+21 x^2+15 x+16$
• $y^2=87 x^6+76 x^5+81 x^4+69 x^3+8 x^2+75 x+80$
• $y^2=25 x^6+65 x^5+52 x^4+41 x^3+66 x^2+66 x+9$
• $y^2=28 x^6+34 x^5+66 x^4+11 x^3+39 x^2+39 x+45$
• $y^2=73 x^6+58 x^5+49 x^4+31 x^3+48 x^2+86 x+73$
• $y^2=74 x^6+96 x^5+51 x^4+58 x^3+46 x^2+42 x+74$
• $y^2=78 x^6+49 x^5+3 x^4+88 x^3+34 x^2+13 x+82$
• $y^2=2 x^6+51 x^5+15 x^4+52 x^3+73 x^2+65 x+22$
• $y^2=5 x^6+12 x^5+77 x^4+53 x^3+79 x^2+60 x+13$
• $y^2=25 x^6+60 x^5+94 x^4+71 x^3+7 x^2+9 x+65$
• $y^2=87 x^6+52 x^5+88 x^4+40 x^3+69 x^2+46 x+48$
• $y^2=47 x^6+66 x^5+52 x^4+6 x^3+54 x^2+36 x+46$
• $y^2=76 x^6+55 x^5+x^4+60 x^3+65 x^2+62 x+73$
• $y^2=89 x^6+81 x^5+5 x^4+9 x^3+34 x^2+19 x+74$
• $y^2=11 x^6+55 x^5+19 x^4+73 x^3+49 x^2+34 x+90$
• $y^2=55 x^6+81 x^5+95 x^4+74 x^3+51 x^2+73 x+62$
• and

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

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 \(\Q(\sqrt{-2}, \sqrt{-185})\).

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

The base change of $A$ to $\F_{97^{2}}$ is 1.9409.agu^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-185}) \)$)$

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.a_gu	$4$	(not in LMFDB)
2.97.ag_s	$8$	(not in LMFDB)
2.97.g_s	$8$	(not in LMFDB)