Abelian variety isogeny class 2.67.i_ad over $\F_{67}$

• Label: 2.67.i_ad
• Base field: $\F_{67}$
• 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:	$\F_{67}$
Dimension:	$2$
L-polynomial:	$1 + 8 x - 3 x^{2} + 536 x^{3} + 4489 x^{4}$
Frobenius angles:	$\pm0.329187292860$, $\pm0.995853959527$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{17})\)
Galois group:	$C_2^2$
Jacobians:	$4$
Cyclic group of points:	no
Non-cyclic primes:	$3$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5031$	$19837233$	$91119459600$	$405985497760089$	$1822892900772336471$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$76$	$4420$	$302956$	$20147044$	$1350165916$	$90457182790$	$6060713668708$	$406067653884484$	$27206535051542932$	$1822837803516804100$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which all are hyperelliptic):

• $y^2=x^6+x^3+56$
• $y^2=x^6+2 x^3+55$
• $y^2=x^6+x^3+16$
• $y^2=x^6+x^3+36$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{67^{3}}$.

Endomorphism algebra over $\F_{67}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{17})\).

Endomorphism algebra over $\overline{\F}_{67}$

The base change of $A$ to $\F_{67^{3}}$ is 1.300763.bqe^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-51}) \)$)$

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
2.67.ai_ad	$2$	(not in LMFDB)
2.67.aq_hq	$3$	(not in LMFDB)
2.67.ai_ad	$6$	(not in LMFDB)