Abelian variety isogeny class 2.37.ag_ab over $\F_{37}$

• Label: 2.37.ag_ab
• Base field: $\F_{37}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: no
• Contains a Jacobian: no

Invariants

Base field:	$\F_{37}$
Dimension:	$2$
L-polynomial:	$1 - 6 x - x^{2} - 222 x^{3} + 1369 x^{4}$
Frobenius angles:	$\pm0.00249485507010$, $\pm0.669161521737$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-7})\)
Galois group:	$C_2^2$
Jacobians:	$0$
Isomorphism classes:	6

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

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1141$	$1822177$	$2520441616$	$3510054088569$	$4807968191624101$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$32$	$1332$	$49754$	$1872868$	$69335072$	$2565524022$	$94931598752$	$3512477380036$	$129961694308178$	$4808584312690932$

Jacobians and polarizations

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{37^{3}}$.

Endomorphism algebra over $\F_{37}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-7})\).

Endomorphism algebra over $\overline{\F}_{37}$

The base change of $A$ to $\F_{37^{3}}$ is 1.50653.ari^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$

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
2.37.g_ab	$2$	(not in LMFDB)
2.37.m_eg	$3$	(not in LMFDB)
2.37.am_eg	$6$	(not in LMFDB)