↑

Invariants

Base field:	$\F_{2}$
Dimension:	$1$
L-polynomial:	$1 - 2 x + 2 x^{2}$
Frobenius angles:	$\pm0.250000000000$
Angle rank:	$0$ (numerical)
Number field:	\(\Q(\sqrt{-1}) \)
Galois group:	$C_2$
Jacobians:	$1$
Isomorphism classes:	1

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1$	$5$	$13$	$25$	$41$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$1$	$5$	$13$	$25$	$41$	$65$	$113$	$225$	$481$	$1025$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobian of 1 curve (which is hyperelliptic):

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{4}}$.

Endomorphism algebra over $\F_{2}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-1}) \).

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

The base change of $A$ to $\F_{2^{4}}$ is the simple isogeny class 1.16.i and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.

Remainder of endomorphism lattice by field

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

  The base change of $A$ to $\F_{2^{2}}$ is the simple isogeny class 1.4.a and its endomorphism algebra is \(\Q(\sqrt{-1}) \).

Base change

This is a primitive isogeny class.

Twists

Additional information

This is the isogeny class of the Jacobian of a function field of class number 1. It also appears as a sporadic example in the classification of abelian varieties with one rational point.

Contact · Citation · Acknowledgments · Editorial Board · Source · SageMath version 9.7 · LMFDB Release 1.2.1