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

• Label: 2.17.ai_bi
• Base field: $\F_{17}$
• Dimension: $2$
• $p$-rank: $1$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{17}$
Dimension:	$2$
L-polynomial:	$( 1 - 8 x + 17 x^{2} )( 1 + 17 x^{2} )$
	$1 - 8 x + 34 x^{2} - 136 x^{3} + 289 x^{4}$
Frobenius angles:	$\pm0.0779791303774$, $\pm0.5$
Angle rank:	$1$ (numerical)
Jacobians:	$16$
Isomorphism classes:	72

This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

$p$-rank:	$1$
Slopes:	$[0, 1/2, 1/2, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$180$	$84240$	$23636340$	$6900940800$	$2014849494900$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$10$	$294$	$4810$	$82622$	$1419050$	$24146406$	$410344490$	$6975653758$	$118588284490$	$2015998927014$

Jacobians and polarizations

This isogeny class contains the Jacobians of 16 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{17^{2}}$.

Endomorphism algebra over $\F_{17}$

The isogeny class factors as 1.17.ai $\times$ 1.17.a and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.17.ai : \(\Q(\sqrt{-1}) \).
• 1.17.a : \(\Q(\sqrt{-17}) \).

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

The base change of $A$ to $\F_{17^{2}}$ is 1.289.abe $\times$ 1.289.bi. The endomorphism algebra for each factor is:

• 1.289.abe : \(\Q(\sqrt{-1}) \).
• 1.289.bi : the quaternion algebra over \(\Q\) ramified at $17$ and $\infty$.

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
2.17.i_bi	$2$	(not in LMFDB)
2.17.ac_bi	$4$	(not in LMFDB)
2.17.c_bi	$4$	(not in LMFDB)