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

• Label: 2.17.am_co
• Base field: $\F_{17}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• 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:	$( 1 - 8 x + 17 x^{2} )( 1 - 4 x + 17 x^{2} )$
	$1 - 12 x + 66 x^{2} - 204 x^{3} + 289 x^{4}$
Frobenius angles:	$\pm0.0779791303774$, $\pm0.338793663197$
Angle rank:	$2$ (numerical)
Jacobians:	$4$
Isomorphism classes:	12
Cyclic group of points:	no
Non-cyclic primes:	$2$

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$140$	$80080$	$24309740$	$6970163200$	$2012913910700$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$6$	$278$	$4950$	$83454$	$1417686$	$24126806$	$410328582$	$6975923326$	$118588965030$	$2015997066518$

Jacobians and polarizations

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

• $y^2=7 x^6+15 x^5+4 x^4+4 x^3+4 x^2+15 x+7$
• $y^2=5 x^6+x^5+16 x^3+x+5$
• $y^2=11 x^6+10 x^5+14 x^4+13 x^3+10 x^2+2 x+1$
• $y^2=3 x^6+5 x^5+x^3+9 x^2+15 x+15$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{17}$

The isogeny class factors as 1.17.ai $\times$ 1.17.ae 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.ae : \(\Q(\sqrt{-13}) \).

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.ae_c	$2$	(not in LMFDB)
2.17.e_c	$2$	(not in LMFDB)
2.17.m_co	$2$	(not in LMFDB)