Properties

Label 2.4.ae_i
Base field $\F_{2^{2}}$
Dimension $2$
$p$-rank $0$
Ordinary no
Supersingular yes
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{2^{2}}$
Dimension:  $2$
L-polynomial:  $( 1 - 2 x )^{2}( 1 + 4 x^{2} )$
  $1 - 4 x + 8 x^{2} - 16 x^{3} + 16 x^{4}$
Frobenius angles:  $0$, $0$, $\pm0.5$
Angle rank:  $0$ (numerical)
Jacobians:  $1$

This isogeny class is not 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:  $[1/2, 1/2, 1/2, 1/2]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $5$ $225$ $3185$ $50625$ $985025$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $1$ $17$ $49$ $193$ $961$ $4097$ $16129$ $64513$ $261121$ $1048577$

Jacobians and polarizations

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{8}}$.

Endomorphism algebra over $\F_{2^{2}}$
The isogeny class factors as 1.4.ae $\times$ 1.4.a and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{2^{2}}$
The base change of $A$ to $\F_{2^{8}}$ is 1.256.abg 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
Remainder of endomorphism lattice by field

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.e_i$2$2.16.a_abg
2.4.c_i$3$2.64.aq_ey
2.4.ai_y$4$2.256.acm_chc

Below is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
2.4.e_i$2$2.16.a_abg
2.4.c_i$3$2.64.aq_ey
2.4.ai_y$4$2.256.acm_chc
2.4.a_ai$4$2.256.acm_chc
2.4.a_i$4$2.256.acm_chc
2.4.i_y$4$2.256.acm_chc
2.4.ac_i$6$(not in LMFDB)
2.4.a_a$8$(not in LMFDB)
2.4.ag_q$12$(not in LMFDB)
2.4.ae_m$12$(not in LMFDB)
2.4.ac_a$12$(not in LMFDB)
2.4.a_ae$12$(not in LMFDB)
2.4.a_e$12$(not in LMFDB)
2.4.c_a$12$(not in LMFDB)
2.4.e_m$12$(not in LMFDB)
2.4.g_q$12$(not in LMFDB)
2.4.ac_e$20$(not in LMFDB)
2.4.c_e$20$(not in LMFDB)