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

• Label: 2.5.af_p
• Base field: $\F_{5}$
• Dimension: $2$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{5}$
Dimension:	$2$
L-polynomial:	$1 - 5 x + 15 x^{2} - 25 x^{3} + 25 x^{4}$
Frobenius angles:	$\pm0.200000000000$, $\pm0.400000000000$
Angle rank:	$0$ (numerical)
Number field:	\(\Q(\zeta_{5})\)
Galois group:	$C_4$
Jacobians:	$2$
Isomorphism classes:	2

This isogeny class is simple but not geometrically simple, primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is supersingular.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$11$	$781$	$19151$	$406901$	$9759376$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$1$	$31$	$151$	$651$	$3126$	$15751$	$78751$	$391251$	$1950001$	$9753126$

Jacobians and polarizations

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

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{5^{10}}$.

Endomorphism algebra over $\F_{5}$

The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{5})\).

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

The base change of $A$ to $\F_{5^{10}}$ is 1.9765625.ajgk^2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $5$ and $\infty$.

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.f_z and its endomorphism algebra is \(\Q(\zeta_{5})\).

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

  The base change of $A$ to $\F_{5^{5}}$ is the simple isogeny class 2.3125.a_ajgk and its endomorphism algebra is the quaternion algebra over \(\Q(\sqrt{5}) \) ramified at both real infinite places.

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.a_ak	$5$	(not in LMFDB)
2.5.f_p	$5$	(not in LMFDB)
2.5.a_f	$15$	(not in LMFDB)