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

• Label: 2.19.am_cp
• 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:	$1 - 12 x + 67 x^{2} - 228 x^{3} + 361 x^{4}$
Frobenius angles:	$\pm0.0409513241932$, $\pm0.374284657527$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{7})\)
Galois group:	$C_2^2$
Jacobians:	$4$
Isomorphism classes:	4

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$189$	$126441$	$47036052$	$16902759321$	$6117677843949$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$8$	$352$	$6860$	$129700$	$2470688$	$47026222$	$893856608$	$16983689284$	$322687697780$	$6131061446752$

Jacobians and polarizations

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

• $y^2=10x^6+7x^5+17x^4+5x^3+14x^2+10x+15$
• $y^2=2x^6+2x^3+4$
• $y^2=x^6+x^3+14$
• $y^2=13x^6+13x^5+14x^4+6x^3+9x^2+3x+13$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{19^{6}}$.

Endomorphism algebra over $\F_{19}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{7})\).

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

The base change of $A$ to $\F_{19^{6}}$ is 1.47045881.aooc^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-21}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{19^{2}}$

  The base change of $A$ to $\F_{19^{2}}$ is the simple isogeny class 2.361.ak_akb and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{7})\).

• Endomorphism algebra over $\F_{19^{3}}$

  The base change of $A$ to $\F_{19^{3}}$ is the simple isogeny class 2.6859.a_aooc and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{7})\).

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.a_k	$3$	(not in LMFDB)
2.19.m_cp	$3$	(not in LMFDB)
2.19.a_k	$6$	(not in LMFDB)