Abelian variety isogeny class 2.61.abb_ls over $\F_{61}$

• Label: 2.61.abb_ls
• Base field: $\F_{61}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{61}$
Dimension:	$2$
L-polynomial:	$( 1 - 14 x + 61 x^{2} )( 1 - 13 x + 61 x^{2} )$
	$1 - 27 x + 304 x^{2} - 1647 x^{3} + 3721 x^{4}$
Frobenius angles:	$\pm0.146275019398$, $\pm0.187058313935$
Angle rank:	$1$ (numerical)
Jacobians:	$0$
Isomorphism classes:	12

This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2352$	$13406400$	$51520795200$	$191807027193600$	$713423619261382512$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$35$	$3601$	$226982$	$13853041$	$844691855$	$51521216038$	$3142748369915$	$191707337131201$	$11694146092834142$	$713342910308616601$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{61^{6}}$.

Endomorphism algebra over $\F_{61}$

The isogeny class factors as 1.61.ao $\times$ 1.61.an and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.61.ao : \(\Q(\sqrt{-3}) \).
• 1.61.an : \(\Q(\sqrt{-3}) \).

Endomorphism algebra over $\overline{\F}_{61}$

The base change of $A$ to $\F_{61^{6}}$ is 1.51520374361.xyoc^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{61^{2}}$
  The base change of $A$ to $\F_{61^{2}}$ is 1.3721.acw $\times$ 1.3721.abv. The endomorphism algebra for each factor is:

  • 1.3721.acw : \(\Q(\sqrt{-3}) \).
  • 1.3721.abv : \(\Q(\sqrt{-3}) \).
• Endomorphism algebra over $\F_{61^{3}}$
  The base change of $A$ to $\F_{61^{3}}$ is 1.226981.aha $\times$ 1.226981.ha. The endomorphism algebra for each factor is:

  • 1.226981.aha : \(\Q(\sqrt{-3}) \).
  • 1.226981.ha : \(\Q(\sqrt{-3}) \).

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
2.61.ab_aci	$2$	(not in LMFDB)
2.61.b_aci	$2$	(not in LMFDB)
2.61.bb_ls	$2$	(not in LMFDB)
2.61.ap_fg	$3$	(not in LMFDB)
2.61.am_ef	$3$	(not in LMFDB)
2.61.a_acw	$3$	(not in LMFDB)
2.61.a_abv	$3$	(not in LMFDB)
2.61.a_er	$3$	(not in LMFDB)
2.61.m_ef	$3$	(not in LMFDB)
2.61.p_fg	$3$	(not in LMFDB)
2.61.bb_ls	$3$	(not in LMFDB)