Abelian variety isogeny class 1.431.a over $\F_{431}$

• Label: 1.431.a
• Base field: $\F_{431}$
• Dimension: $1$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{431}$
Dimension:	$1$
L-polynomial:	$1 + 431 x^{2}$
Frobenius angles:	$\pm0.5$
Angle rank:	$0$ (numerical)
Number field:	\(\Q(\sqrt{-431}) \)
Galois group:	$C_2$
Jacobians:	$42$
Isomorphism classes:	42

This isogeny class is simple and 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:	$[1/2, 1/2]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$432$	$186624$	$80062992$	$34506777600$	$14872581271152$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$432$	$186624$	$80062992$	$34506777600$	$14872581271152$	$6410082687992064$	$2762745569510280912$	$1190743340389916774400$	$513210379737799292308272$	$221193673667021240147407104$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 42 curves (of which 0 are hyperelliptic):

• $y^2=x^3+286 x+286$
• $y^2=x^3+410 x+284$
• $y^2=x^3+72 x+73$
• $y^2=x^3+114 x+114$
• $y^2=x^3+369 x+369$
• $y^2=x^3+205 x+205$
• $y^2=x^3+67 x+38$
• $y^2=x^3+13 x+13$
• $y^2=x^3+317 x+317$
• $y^2=x^3+156 x+230$
• $y^2=x^3+206 x+149$
• $y^2=x^3+151 x+151$
• $y^2=x^3+103 x+290$
• $y^2=x^3+84 x+84$
• $y^2=x^3+76 x+76$
• $y^2=x^3+7 x$
• $y^2=x^3+290 x+306$
• $y^2=x^3+26 x+182$
• $y^2=x^3+237 x+366$
• $y^2=x^3+222 x+222$
• and

• $y^2=x^3+426 x+426$
• $y^2=x^3+x$
• $y^2=x^3+414 x+312$
• $y^2=x^3+63 x+10$
• $y^2=x^3+276 x+208$
• $y^2=x^3+186 x+9$
• $y^2=x^3+144 x+146$
• $y^2=x^3+7$
• $y^2=x^3+39 x+39$
• $y^2=x^3+17 x+119$
• $y^2=x^3+186 x+186$
• $y^2=x^3+282 x+282$
• $y^2=x^3+362 x+362$
• $y^2=x^3+302 x+302$
• $y^2=x^3+222 x+261$
• $y^2=x^3+261 x+261$
• $y^2=x^3+399 x+399$
• $y^2=x^3+377 x+53$
• $y^2=x^3+210 x+210$
• $y^2=x^3+187 x+16$
• $y^2=x^3+132 x+62$
• $y^2=x^3+1$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{431^{2}}$.

Endomorphism algebra over $\F_{431}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-431}) \).

Endomorphism algebra over $\overline{\F}_{431}$

The base change of $A$ to $\F_{431^{2}}$ is the simple isogeny class 1.185761.bhe and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $431$ and $\infty$.

Base change

This is a primitive isogeny class.

Twists

This isogeny class has no twists.