Properties

Label 2.2.a_ae
Base field $\F_{2}$
Dimension $2$
$p$-rank $0$
Ordinary no
Supersingular yes
Simple yes
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian no

Learn more

Invariants

Base field:  $\F_{2}$
Dimension:  $2$
L-polynomial:  $( 1 - 2 x^{2} )^{2}$
  $1 - 4x^{2} + 4x^{4}$
Frobenius angles:  $0$, $0$, $1$, $1$
Angle rank:  $0$ (numerical)
Number field:  \(\Q(\sqrt{2}) \)
Galois group:  $C_2$
Jacobians:  0

This isogeny class is simple but not geometrically simple.

Newton polygon

This isogeny class is supersingular.

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

Point counts

This isogeny class is principally polarizable, but does not contain a Jacobian.

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $1$ $1$ $49$ $81$ $961$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $3$ $-3$ $9$ $1$ $33$ $33$ $129$ $193$ $513$ $897$

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
The endomorphism algebra of this simple isogeny class is the quaternion algebra over \(\Q(\sqrt{2}) \) ramified at both real infinite places.
Endomorphism algebra over $\overline{\F}_{2}$
The base change of $A$ to $\F_{2^{2}}$ is 1.4.ae 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
All geometric endomorphisms are defined over $\F_{2^{2}}$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension degreeCommon base change
2.2.a_c$3$2.8.a_aq
2.2.a_e$4$2.16.aq_ds
2.2.a_c$6$2.64.abg_ou
Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
2.2.a_c$3$2.8.a_aq
2.2.a_e$4$2.16.aq_ds
2.2.a_c$6$2.64.abg_ou
2.2.ae_i$8$2.256.acm_chc
2.2.ac_e$8$2.256.acm_chc
2.2.a_a$8$2.256.acm_chc
2.2.c_e$8$2.256.acm_chc
2.2.e_i$8$2.256.acm_chc
2.2.a_ac$12$(not in LMFDB)
2.2.ac_c$24$(not in LMFDB)
2.2.c_c$24$(not in LMFDB)

Additional information

This isogeny class appears as a sporadic example in the classification of abelian varieties with one rational point.