Abelian variety isogeny class 2.11.aj_bm over $\F_{11}$

• Label: 2.11.aj_bm
• Base field: $\F_{11}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{11}$
Dimension:	$2$
L-polynomial:	$1 - 9 x + 38 x^{2} - 99 x^{3} + 121 x^{4}$
Frobenius angles:	$\pm0.0468428922585$, $\pm0.380176225592$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{17})\)
Galois group:	$C_2^2$
Jacobians:	$2$
Isomorphism classes:	2

This isogeny class is simple but not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$52$	$13936$	$1769872$	$211214016$	$25719331612$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$3$	$117$	$1332$	$14425$	$159693$	$1768182$	$19486911$	$214376689$	$2357947692$	$25937131077$

Jacobians and polarizations

This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=7x^6+8x^5+9x^4+4x^3+8x^2+x+7$
• $y^2=8x^6+2x^5+8x^4+2x^2+3x+5$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{11^{6}}$.

Endomorphism algebra over $\F_{11}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{17})\).

Endomorphism algebra over $\overline{\F}_{11}$

The base change of $A$ to $\F_{11^{6}}$ is 1.1771561.acna^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-51}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{11^{2}}$

  The base change of $A$ to $\F_{11^{2}}$ is the simple isogeny class 2.121.af_ads and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{17})\).

• Endomorphism algebra over $\F_{11^{3}}$

  The base change of $A$ to $\F_{11^{3}}$ is the simple isogeny class 2.1331.a_acna and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{17})\).

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
2.11.a_f	$3$	(not in LMFDB)
2.11.j_bm	$3$	(not in LMFDB)
2.11.a_f	$6$	(not in LMFDB)