Abelian variety isogeny class 2.49.az_jt over $\F_{7^{2}}$

• Label: 2.49.az_jt
• Base field: $\F_{7^{2}}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{7^{2}}$
Dimension:	$2$
L-polynomial:	$1 - 25 x + 253 x^{2} - 1225 x^{3} + 2401 x^{4}$
Frobenius angles:	$\pm0.0745256647597$, $\pm0.197834195333$
Angle rank:	$2$ (numerical)
Number field:	4.0.17525.1
Galois group:	$D_{4}$
Jacobians:	$3$
Isomorphism classes:	3

This isogeny class is simple and 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1405$	$5486525$	$13803182245$	$33238471241525$	$79798057165282000$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$25$	$2283$	$117325$	$5765763$	$282495750$	$13841444163$	$678223767925$	$33232931040483$	$1628413576885225$	$79792266127451998$

Jacobians and polarizations

This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=ax^6+(a+6)x^5+(3a+5)x^3+(6a+5)x^2+(3a+6)x+3a+2$
• $y^2=(3a+3)x^6+(4a+2)x^5+(a+5)x^4+(4a+4)x^3+(4a+3)x^2+x+3a+3$
• $y^2=(5a+5)x^6+(2a+2)x^5+(2a+5)x^4+(2a+1)x^3+4ax^2+(3a+4)x+2a+3$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{7^{2}}$.

Endomorphism algebra over $\F_{7^{2}}$

The endomorphism algebra of this simple isogeny class is 4.0.17525.1.

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
2.49.z_jt	$2$	(not in LMFDB)