Properties

Label 2.4.a_ai
Base Field $\F_{2^{2}}$
Dimension $2$
Ordinary No
$p$-rank $0$
Principally polarizable Yes
Contains a Jacobian No

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Invariants

Base field:  $\F_{2^{2}}$
Dimension:  $2$
L-polynomial:  $( 1 - 2 x )^{2}( 1 + 2 x )^{2}$
Frobenius angles:  $0$, $0$, $1$, $1$
Angle rank:  $0$ (numerical)
Jacobians:  0

This isogeny class is not 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 6 7 8 9 10
$A(\F_{q^r})$ 9 81 3969 50625 1046529 15752961 268402689 4228250625 68718952449 1095222947841

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 5 1 65 193 1025 3841 16385 64513 262145 1044481

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
The isogeny class factors as 1.4.ae $\times$ 1.4.e and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
  • 1.4.ae : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 1.4.e : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
Endomorphism algebra over $\overline{\F}_{2^{2}}$
The base change of $A$ to $\F_{2^{4}}$ is 1.16.ai 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 $\F_{2^{4}}$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.4.ai_y$2$2.16.aq_ds
2.4.i_y$2$2.16.aq_ds
2.4.ag_q$3$2.64.a_aey
2.4.a_e$3$2.64.a_aey
2.4.g_q$3$2.64.a_aey
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.4.ai_y$2$2.16.aq_ds
2.4.i_y$2$2.16.aq_ds
2.4.ag_q$3$2.64.a_aey
2.4.a_e$3$2.64.a_aey
2.4.g_q$3$2.64.a_aey
2.4.ae_i$4$2.256.acm_chc
2.4.a_i$4$2.256.acm_chc
2.4.e_i$4$2.256.acm_chc
2.4.ae_m$6$(not in LMFDB)
2.4.ac_a$6$(not in LMFDB)
2.4.c_a$6$(not in LMFDB)
2.4.e_m$6$(not in LMFDB)
2.4.a_a$8$(not in LMFDB)
2.4.ac_e$10$(not in LMFDB)
2.4.c_e$10$(not in LMFDB)
2.4.ac_i$12$(not in LMFDB)
2.4.a_ae$12$(not in LMFDB)
2.4.c_i$12$(not in LMFDB)