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

• Label: 2.49.ax_iu
• Base field: $\F_{7^{2}}$
• 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:	$\F_{7^{2}}$
Dimension:	$2$
L-polynomial:	$( 1 - 13 x + 49 x^{2} )( 1 - 10 x + 49 x^{2} )$
	$1 - 23 x + 228 x^{2} - 1127 x^{3} + 2401 x^{4}$
Frobenius angles:	$\pm0.121037718324$, $\pm0.246751714429$
Angle rank:	$2$ (numerical)
Jacobians:	$9$
Isomorphism classes:	48

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1480$	$5594400$	$13863035680$	$33259222684800$	$79802395849731400$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$27$	$2329$	$117834$	$5769361$	$282511107$	$13841455102$	$678223455363$	$33232930550881$	$1628413613397546$	$79792266686047849$

Jacobians and polarizations

This isogeny class contains the Jacobians of 9 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

Decomposition and endomorphism algebra

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

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

The isogeny class factors as 1.49.an $\times$ 1.49.ak and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.49.an : \(\Q(\sqrt{-3}) \).
• 1.49.ak : \(\Q(\sqrt{-6}) \).

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
2.49.ad_abg	$2$	(not in LMFDB)
2.49.d_abg	$2$	(not in LMFDB)
2.49.x_iu	$2$	(not in LMFDB)
2.49.ai_da	$3$	(not in LMFDB)
2.49.b_am	$3$	(not in LMFDB)