Properties

Label 2.4.ai_y
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:  $( 1 - 2 x )^{4}$
Frobenius angles:  $0$, $0$, $0$, $0$
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
$A(\F_{q^r})$ 1 81 2401 50625 923521 15752961 260144641 4228250625 68184176641 1095222947841

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -3 1 33 193 897 3841 15873 64513 260097 1044481

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
The isogeny class factors as 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 isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{2}}$.

SubfieldPrimitive Model
$\F_{2}$2.2.a_ae

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.4.a_ai$2$2.16.aq_ds
2.4.i_y$2$2.16.aq_ds
2.4.ac_a$3$2.64.abg_ou
2.4.e_m$3$2.64.abg_ou
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.4.a_ai$2$2.16.aq_ds
2.4.i_y$2$2.16.aq_ds
2.4.ac_a$3$2.64.abg_ou
2.4.e_m$3$2.64.abg_ou
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.c_e$5$2.1024.aey_jci
2.4.ag_q$6$(not in LMFDB)
2.4.ae_m$6$(not in LMFDB)
2.4.a_e$6$(not in LMFDB)
2.4.c_a$6$(not in LMFDB)
2.4.g_q$6$(not in LMFDB)
2.4.a_a$8$(not in LMFDB)
2.4.ac_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)