Abelian variety isogeny class 2.17.am_cn over $\F_{17}$

• Label: 2.17.am_cn
• Base field: $\F_{17}$
• 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:	$\F_{17}$
Dimension:	$2$
L-polynomial:	$1 - 12 x + 65 x^{2} - 204 x^{3} + 289 x^{4}$
Frobenius angles:	$\pm0.0157896134134$, $\pm0.349122946747$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{5})\)
Galois group:	$C_2^2$
Jacobians:	$2$
Isomorphism classes:	2

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$139$	$79369$	$24128176$	$6943914441$	$2010360903259$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$6$	$276$	$4914$	$83140$	$1415886$	$24118782$	$410293470$	$6975736324$	$118587876498$	$2015991481236$

Jacobians and polarizations

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

• $y^2=3x^6+2x^5+13x^4+12x^3+14x^2+16x+3$
• $y^2=7x^6+10x^5+3x^3+4x^2+15x+7$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{17^{6}}$.

Endomorphism algebra over $\F_{17}$

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

Endomorphism algebra over $\overline{\F}_{17}$

The base change of $A$ to $\F_{17^{6}}$ is 1.24137569.anxi^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{17^{2}}$

  The base change of $A$ to $\F_{17^{2}}$ is the simple isogeny class 2.289.ao_adp and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{5})\).

• Endomorphism algebra over $\F_{17^{3}}$

  The base change of $A$ to $\F_{17^{3}}$ is the simple isogeny class 2.4913.a_anxi and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{5})\).

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
2.17.m_cn	$2$	(not in LMFDB)
2.17.a_o	$3$	(not in LMFDB)
2.17.m_cn	$3$	(not in LMFDB)