Abelian variety isogeny class 2.5.ad_e over $\F_{5}$

• Label: 2.5.ad_e
• Base field: $\F_{5}$
• 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:	$\F_{5}$
Dimension:	$2$
L-polynomial:	$1 - 3 x + 4 x^{2} - 15 x^{3} + 25 x^{4}$
Frobenius angles:	$\pm0.0673911931187$, $\pm0.599275473548$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-11})\)
Galois group:	$C_2^2$
Jacobians:	$3$
Isomorphism classes:	3

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$12$	$576$	$11664$	$361728$	$9943932$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$3$	$25$	$90$	$577$	$3183$	$15478$	$77619$	$391777$	$1954962$	$9762625$

Jacobians and polarizations

This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{5^{3}}$.

Endomorphism algebra over $\F_{5}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-11})\).

Endomorphism algebra over $\overline{\F}_{5}$

The base change of $A$ to $\F_{5^{3}}$ is 1.125.as^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$

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
2.5.d_e	$2$	2.25.ab_ay
2.5.g_t	$3$	2.125.abk_wc
2.5.ag_t	$6$	(not in LMFDB)