Abelian variety isogeny class 2.47.e_abf over $\F_{47}$

• Label: 2.47.e_abf
• Base field: $\F_{47}$
• 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:	$\F_{47}$
Dimension:	$2$
L-polynomial:	$1 + 4 x - 31 x^{2} + 188 x^{3} + 2209 x^{4}$
Frobenius angles:	$\pm0.260897533342$, $\pm0.927564200009$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-43})\)
Galois group:	$C_2^2$
Jacobians:	$0$
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2371$	$4711177$	$10883496976$	$23819414107849$	$52606050879173251$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$52$	$2132$	$104824$	$4881348$	$229375172$	$10779130622$	$506621872172$	$23811279677956$	$1119130411633768$	$52599132687006932$

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 $\F_{47^{3}}$.

Endomorphism algebra over $\F_{47}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-43})\).

Endomorphism algebra over $\overline{\F}_{47}$

The base change of $A$ to $\F_{47^{3}}$ is 1.103823.tg^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
2.47.ae_abf	$2$	(not in LMFDB)
2.47.ai_eg	$3$	(not in LMFDB)
2.47.ae_abf	$6$	(not in LMFDB)