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

• Label: 2.5.d_i
• 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:	$1 + 3 x + 8 x^{2} + 15 x^{3} + 25 x^{4}$
Frobenius angles:	$\pm0.459924988887$, $\pm0.793258322220$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{17})\)
Galois group:	$C_2^2$
Jacobians:	$4$
Isomorphism classes:	4
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$52$	$832$	$15808$	$389376$	$9225892$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$9$	$33$	$126$	$625$	$2949$	$15990$	$78465$	$389377$	$1953126$	$9761433$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which all are hyperelliptic):

• $y^2=4 x^6+3 x^5+2 x^4+3 x^3+2 x+2$
• $y^2=x^6+4 x^5+3 x^4+x^3+3 x^2+2 x+1$
• $y^2=x^5+4 x^4+3 x^2+x+4$
• $y^2=x^6+4 x^5+3 x^3+3 x+4$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{5^{6}}$.

Endomorphism algebra over $\F_{5}$

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

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

The base change of $A$ to $\F_{5^{6}}$ is 1.15625.ha^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-51}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{5^{2}}$

  The base change of $A$ to $\F_{5^{2}}$ is the simple isogeny class 2.25.h_y and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{17})\).

• Endomorphism algebra over $\F_{5^{3}}$

  The base change of $A$ to $\F_{5^{3}}$ is the simple isogeny class 2.125.a_ha 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
2.5.ad_i	$2$	2.25.h_y
2.5.ad_i	$3$	2.125.a_ha
2.5.a_ah	$3$	2.125.a_ha