Abelian variety isogeny class 2.19.e_ad over $\F_{19}$

• Label: 2.19.e_ad
• Base field: $\F_{19}$
• 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:	$\F_{19}$
Dimension:	$2$
L-polynomial:	$1 + 4 x - 3 x^{2} + 76 x^{3} + 361 x^{4}$
Frobenius angles:	$\pm0.318398499578$, $\pm0.985065166245$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{5})\)
Galois group:	$C_2^2$
Jacobians:	$2$
Isomorphism classes:	6
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$439$	$122481$	$49336576$	$16952472729$	$6136426744999$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$24$	$340$	$7188$	$130084$	$2478264$	$47019526$	$893883336$	$16983359044$	$322689770412$	$6131065988500$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which all are hyperelliptic):

• $y^2=2 x^6+2 x^3+13$
• $y^2=2 x^6+4 x^3+8$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{19^{3}}$.

Endomorphism algebra over $\F_{19}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{5})\).

Endomorphism algebra over $\overline{\F}_{19}$

The base change of $A$ to $\F_{19^{3}}$ is 1.6859.gi^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$

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
2.19.ae_ad	$2$	(not in LMFDB)
2.19.ai_cc	$3$	(not in LMFDB)
2.19.ae_ad	$6$	(not in LMFDB)