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

• Label: 2.49.ay_jh
• 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:	$( 1 - 13 x + 49 x^{2} )( 1 - 11 x + 49 x^{2} )$
	$1 - 24 x + 241 x^{2} - 1176 x^{3} + 2401 x^{4}$
Frobenius angles:	$\pm0.121037718324$, $\pm0.212295615010$
Angle rank:	$1$ (numerical)
Jacobians:	$10$
Isomorphism classes:	16

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1443$	$5545449$	$13841440704$	$33256196289225$	$79804666755947523$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$26$	$2308$	$117650$	$5768836$	$282519146$	$13841594206$	$678224605994$	$33232935313156$	$1628413597910450$	$79792266220276228$

Jacobians and polarizations

This isogeny class contains the Jacobians of 10 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{7^{12}}$.

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

The isogeny class factors as 1.49.an $\times$ 1.49.al 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.al : \(\Q(\sqrt{-3}) \).

Endomorphism algebra over $\overline{\F}_{7^{2}}$

The base change of $A$ to $\F_{7^{12}}$ is 1.13841287201.itby^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{7^{4}}$
  The base change of $A$ to $\F_{7^{4}}$ is 1.2401.act $\times$ 1.2401.ax. The endomorphism algebra for each factor is:

  • 1.2401.act : \(\Q(\sqrt{-3}) \).
  • 1.2401.ax : \(\Q(\sqrt{-3}) \).
• Endomorphism algebra over $\F_{7^{6}}$
  The base change of $A$ to $\F_{7^{6}}$ is 1.117649.ala $\times$ 1.117649.la. The endomorphism algebra for each factor is:

  • 1.117649.ala : \(\Q(\sqrt{-3}) \).
  • 1.117649.la : \(\Q(\sqrt{-3}) \).

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.ac_abt	$2$	(not in LMFDB)
2.49.c_abt	$2$	(not in LMFDB)
2.49.y_jh	$2$	(not in LMFDB)
2.49.ap_eu	$3$	(not in LMFDB)
2.49.aj_cy	$3$	(not in LMFDB)
2.49.a_act	$3$	(not in LMFDB)
2.49.a_ax	$3$	(not in LMFDB)
2.49.a_dq	$3$	(not in LMFDB)
2.49.j_cy	$3$	(not in LMFDB)
2.49.p_eu	$3$	(not in LMFDB)
2.49.y_jh	$3$	(not in LMFDB)