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

• Label: 2.97.a_ahi
• 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:	$1 - 190 x^{2} + 9409 x^{4}$
Frobenius angles:	$\pm0.0323752631788$, $\pm0.967624736821$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(i, \sqrt{6})\)
Galois group:	$C_2^2$
Jacobians:	$21$
Isomorphism classes:	72
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$9220$	$85008400$	$832970509060$	$7834374144000000$	$73742412680461764100$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$98$	$9030$	$912674$	$88494718$	$8587340258$	$832969013190$	$80798284478114$	$7837433351159038$	$760231058654565218$	$73742412671430702150$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 21 curves (of which all are hyperelliptic):

• $y^2=31 x^6+55 x^5+95 x^4+62 x^2+10 x+16$
• $y^2=27 x^6+91 x^5+32 x^4+39 x^3+92 x^2+8 x+42$
• $y^2=20 x^6+5 x^5+34 x^4+76 x^2+21 x+69$
• $y^2=61 x^6+4 x^5+25 x^4+32 x^3+39 x^2+49 x+57$
• $y^2=18 x^6+95 x^5+47 x^4+66 x^3+67 x^2+73 x+52$
• $y^2=10 x^6+64 x^5+2 x^4+44 x^3+28 x^2+31 x+86$
• $y^2=39 x^6+37 x^5+21 x^4+63 x^2+55 x+83$
• $y^2=59 x^6+36 x^5+88 x^4+x^3+41 x^2+25 x+31$
• $y^2=41 x^6+92 x^5+49 x^4+8 x^2+19 x+29$
• $y^2=7 x^6+78 x^5+85 x^4+27 x^2+78 x+40$
• $y^2=26 x^6+84 x^5+3 x^4+85 x^2+14 x+82$
• $y^2=78 x^6+42 x^5+22 x^4+16 x^2+82 x+19$
• $y^2=87 x^6+55 x^5+29 x^4+63 x^2+59 x+23$
• $y^2=7 x^6+20 x^5+86 x^4+35 x^3+59 x^2+7 x+94$
• $y^2=x^6+61 x^5+61 x^4+12 x^3+52 x^2+65 x+55$
• $y^2=87 x^6+70 x^5+26 x^4+15 x^2+12 x+82$
• $y^2=78 x^6+10 x^5+18 x^4+54 x^2+7 x+69$
• $y^2=4 x^6+28 x^5+9 x^4+66 x^2+57 x+65$
• $y^2=60 x^6+28 x^5+24 x^4+6 x^2+71 x+7$
• $y^2=18 x^6+35 x^5+82 x^4+63 x^3+22 x^2+2 x+19$
• $y^2=61 x^6+44 x^5+4 x^4+62 x^2+2 x+43$

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 \(\Q(i, \sqrt{6})\).

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

The base change of $A$ to $\F_{97^{2}}$ is 1.9409.ahi^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-6}) \)$)$

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.ae_hq	$4$	(not in LMFDB)
2.97.a_hi	$4$	(not in LMFDB)
2.97.e_hq	$4$	(not in LMFDB)