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

• Label: 2.97.a_ahl
• 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: no

Invariants

Base field:	$\F_{97}$
Dimension:	$2$
L-polynomial:	$1 - 193 x^{2} + 9409 x^{4}$
Frobenius angles:	$\pm0.0161666856435$, $\pm0.983833314357$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(i, \sqrt{43})\)
Galois group:	$C_2^2$
Jacobians:	$0$

This isogeny class is simple but not geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable.

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$9217$	$84953089$	$832970263684$	$7834170744744201$	$73742412674486072257$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$98$	$9024$	$912674$	$88492420$	$8587340258$	$832968522438$	$80798284478114$	$7837433269090564$	$760231058654565218$	$73742412659479318464$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

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 \(\Q(i, \sqrt{43})\).

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

The base change of $A$ to $\F_{97^{2}}$ is 1.9409.ahl^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-43}) \)$)$

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.ac_hn	$4$	(not in LMFDB)
2.97.a_hl	$4$	(not in LMFDB)
2.97.c_hn	$4$	(not in LMFDB)