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

• Label: 2.11.d_o
• 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:	$1 + 3 x + 14 x^{2} + 33 x^{3} + 121 x^{4}$
Frobenius angles:	$\pm0.417424003664$, $\pm0.750757336997$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{41})\)
Galois group:	$C_2^2$
Jacobians:	$10$
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$172$	$17200$	$1771600$	$216100800$	$25721253412$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$15$	$141$	$1332$	$14761$	$159705$	$1771638$	$19501875$	$214343761$	$2357947692$	$25937141901$

Jacobians and polarizations

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

• $y^2=7 x^6+5 x^5+9 x^4+9 x^3+6 x^2+6 x+1$
• $y^2=8 x^6+6 x^5+7 x^4+10 x^3+6 x^2+9 x+3$
• $y^2=4 x^6+x^5+x^4+7 x^2+6 x+9$
• $y^2=9 x^6+2 x^3+3 x^2+10 x+9$
• $y^2=5 x^6+7 x^5+6 x^4+7 x^3+2 x^2+3 x+5$
• $y^2=4 x^6+3 x^5+6 x^4+4 x^3+4 x^2+6 x+7$
• $y^2=9 x^6+6 x^5+x^4+5 x^3+5 x^2+9 x+5$
• $y^2=9 x^6+3 x^5+10 x^4+8 x^3+3 x^2+10 x+10$
• $y^2=8 x^6+6 x^5+6 x^4+6 x^3+2 x^2+10 x$
• $y^2=5 x^6+4 x^5+x^4+2 x^3+9 x^2+9 x+5$

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 \(\Q(\sqrt{-3}, \sqrt{41})\).

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

The base change of $A$ to $\F_{11^{6}}$ is 1.1771561.bm^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-123}) \)$)$

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.t_jg and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{41})\).

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

  The base change of $A$ to $\F_{11^{3}}$ is the simple isogeny class 2.1331.a_bm and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{41})\).

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.ad_o	$3$	(not in LMFDB)
2.11.a_at	$3$	(not in LMFDB)
2.11.a_at	$6$	(not in LMFDB)