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

• Label: 2.19.am_ct
• Base field: $\F_{19}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{19}$
Dimension:	$2$
L-polynomial:	$1 - 12 x + 71 x^{2} - 228 x^{3} + 361 x^{4}$
Frobenius angles:	$\pm0.152835520717$, $\pm0.337154730080$
Angle rank:	$2$ (numerical)
Number field:	4.0.134928.2
Galois group:	$D_{4}$
Jacobians:	$2$
Isomorphism classes:	2

This isogeny class is simple and 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$193$	$129889$	$48043876$	$17059231593$	$6132523129513$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$8$	$360$	$7004$	$130900$	$2476688$	$47045454$	$893904656$	$16983913828$	$322689260324$	$6131067761880$

Jacobians and polarizations

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

• $y^2=13x^6+15x^5+10x^4+x^3+12x^2+2$
• $y^2=9x^6+10x^5+12x^4+12x^3+16x^2+3x+3$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{19}$

The endomorphism algebra of this simple isogeny class is 4.0.134928.2.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.19.m_ct	$2$	(not in LMFDB)