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

• Label: 2.17.aj_bs
• 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:	$1 - 9 x + 44 x^{2} - 153 x^{3} + 289 x^{4}$
Frobenius angles:	$\pm0.116336544503$, $\pm0.449669877836$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{41})\)
Galois group:	$C_2^2$
Jacobians:	$8$
Isomorphism classes:	8
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$172$	$85312$	$24143296$	$6931770624$	$2014449490252$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$9$	$297$	$4914$	$82993$	$1418769$	$24149022$	$410408721$	$6975870241$	$118587876498$	$2015996344857$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):

• $y^2=14 x^6+11 x^5+2 x^4+8 x^3+4 x^2+14 x+13$
• $y^2=5 x^6+4 x^5+7 x^4+3 x^3+13 x^2+6 x$
• $y^2=14 x^6+2 x^5+15 x^4+2 x^3+x^2+14 x+5$
• $y^2=9 x^5+7 x^4+10 x^3+12 x^2+7 x+12$
• $y^2=7 x^6+6 x^5+8 x^3+12 x^2+10$
• $y^2=11 x^6+8 x^5+3 x^4+8 x^3+9 x^2+7 x+11$
• $y^2=6 x^6+15 x^5+9 x^4+16 x^3+6 x^2+14 x+13$
• $y^2=6 x^6+13 x^5+16 x^4+2 x^3+4 x^2+16 x+12$

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 \(\Q(\sqrt{-3}, \sqrt{41})\).

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

The base change of $A$ to $\F_{17^{6}}$ is 1.24137569.img^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-123}) \)$)$

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.h_ajg and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{41})\).

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

  The base change of $A$ to $\F_{17^{3}}$ is the simple isogeny class 2.4913.a_img and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{41})\).

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.j_bs	$2$	(not in LMFDB)
2.17.a_ah	$3$	(not in LMFDB)
2.17.j_bs	$3$	(not in LMFDB)