Abelian variety isogeny class 2.83.i_at over $\F_{83}$

• Label: 2.83.i_at
• Base field: $\F_{83}$
• 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:	$\F_{83}$
Dimension:	$2$
L-polynomial:	$1 + 8 x - 19 x^{2} + 664 x^{3} + 6889 x^{4}$
Frobenius angles:	$\pm0.311354066930$, $\pm0.978020733597$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-67})\)
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$7543$	$46759057$	$328636199824$	$2252132071686169$	$15516418741621298023$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$92$	$6788$	$574748$	$47454948$	$3939136492$	$326938279718$	$27136051408852$	$2252292148606276$	$186940256673655844$	$15516041188514611268$

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 $\F_{83^{3}}$.

Endomorphism algebra over $\F_{83}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-67})\).

Endomorphism algebra over $\overline{\F}_{83}$

The base change of $A$ to $\F_{83^{3}}$ is 1.571787.cey^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-67}) \)$)$

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
2.83.ai_at	$2$	(not in LMFDB)
2.83.aq_iw	$3$	(not in LMFDB)
2.83.a_dy	$6$	(not in LMFDB)