Abelian variety isogeny class 2.25.am_cv over $\F_{5^{2}}$

• Label: 2.25.am_cv
• Base field: $\F_{5^{2}}$
• 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:	$\F_{5^{2}}$
Dimension:	$2$
L-polynomial:	$1 - 12 x + 73 x^{2} - 300 x^{3} + 625 x^{4}$
Frobenius angles:	$\pm0.0897012299332$, $\pm0.423034563267$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{13})\)
Galois group:	$C_2^2$
Jacobians:	$20$
Isomorphism classes:	20

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$387$	$391257$	$244144368$	$152102332521$	$95300542986867$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$14$	$628$	$15626$	$389380$	$9758774$	$244148110$	$6103731830$	$152588661892$	$3814697265626$	$95367435521908$

Jacobians and polarizations

This isogeny class contains the Jacobians of 20 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=2x^6+(a+4)x^5+2x^4+(2a+2)x^3+(3a+4)x^2+(3a+2)x+a$
• $y^2=x^6+x^3+3a$
• $y^2=3ax^6+(4a+4)x^5+(a+3)x^4+(2a+3)x^3+ax^2+(3a+4)x+1$
• $y^2=(4a+1)x^6+(4a+3)x^5+(a+3)x^4+(3a+1)x^3+(3a+1)x^2+(4a+3)x+4$
• $y^2=ax^6+ax^3+2a+1$
• $y^2=(4a+1)x^6+x^5+(3a+2)x^4+4ax^3+(4a+2)x+4a$
• $y^2=x^6+x^3+2a+3$
• $y^2=4ax^6+(4a+2)x^5+(4a+4)x^4+(4a+4)x^3+(a+1)x^2+3ax+a+3$
• $y^2=(2a+3)x^6+4x^5+(2a+4)x^4+(4a+3)x^3+4ax^2+(4a+1)x+2$
• $y^2=(a+4)x^6+(2a+1)x^5+(3a+2)x^4+(a+1)x^3+3ax^2+2x+4a$
• $y^2=(2a+3)x^6+(2a+3)x^5+(2a+1)x^4+(2a+4)x^3+2ax^2+3ax+4a+1$
• $y^2=(4a+3)x^6+(3a+4)x^5+(4a+2)x^4+4ax^3+(4a+2)x^2+4x+4a+1$
• $y^2=(a+4)x^6+2x^5+(4a+2)x^4+(2a+1)x^3+(4a+3)x^2+(2a+3)x+2a+3$
• $y^2=(a+2)x^6+(a+3)x^5+(3a+4)x^4+ax^3+(4a+1)x^2+2ax+4a$
• $y^2=(4a+4)x^6+(a+2)x^5+3x^4+3ax^3+3ax^2+4ax+3a+3$
• $y^2=x^6+(2a+2)x^5+(4a+3)x^4+ax^3+(2a+4)x^2+(3a+1)x+4a+4$
• $y^2=(3a+2)x^6+2x^5+3x^4+(4a+2)x^3+2ax^2+3x+a$
• $y^2=ax^6+ax^3+4$
• $y^2=ax^6+3ax^5+4ax^4+(4a+4)x^3+(4a+1)x^2+4ax+a+4$
• $y^2=(4a+1)x^6+x^5+(a+3)x^4+(2a+2)x^3+4x^2+a$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{5^{12}}$.

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

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{13})\).

Endomorphism algebra over $\overline{\F}_{5^{2}}$

The base change of $A$ to $\F_{5^{12}}$ is 1.244140625.fny^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-39}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{5^{4}}$

  The base change of $A$ to $\F_{5^{4}}$ is the simple isogeny class 2.625.c_axx and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{13})\).

• Endomorphism algebra over $\F_{5^{6}}$

  The base change of $A$ to $\F_{5^{6}}$ is the simple isogeny class 2.15625.a_fny and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{13})\).

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
2.25.m_cv	$2$	2.625.c_axx
2.25.a_ac	$3$	(not in LMFDB)
2.25.m_cv	$3$	(not in LMFDB)