Properties

Label 2.32.aq_ey
Base field $\F_{2^{5}}$
Dimension $2$
$p$-rank $0$
Ordinary no
Supersingular yes
Simple no
Geometrically simple no
Primitive no
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{2^{5}}$
Dimension:  $2$
L-polynomial:  $( 1 - 8 x + 32 x^{2} )^{2}$
  $1 - 16 x + 128 x^{2} - 512 x^{3} + 1024 x^{4}$
Frobenius angles:  $\pm0.250000000000$, $\pm0.250000000000$
Angle rank:  $0$ (numerical)
Jacobians:  $5$

This isogeny class is not simple, not 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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $625$ $1050625$ $1090650625$ $1103812890625$ $1126449796890625$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $17$ $1025$ $33281$ $1052673$ $33570817$ $1073741825$ $34359214081$ $1099507433473$ $35184355311617$ $1125899906842625$

Jacobians and polarizations

This isogeny class contains the Jacobians of 5 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{20}}$.

Endomorphism algebra over $\F_{2^{5}}$
The isogeny class factors as 1.32.ai 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
Endomorphism algebra over $\overline{\F}_{2^{5}}$
The base change of $A$ to $\F_{2^{20}}$ is 1.1048576.dau 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 isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{5}}$.

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

Twists

Below are some of the twists of this isogeny class.

TwistExtension degreeCommon base change
2.32.a_a$2$2.1024.a_dau
2.32.q_ey$2$2.1024.a_dau
2.32.i_bg$3$(not in LMFDB)

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

TwistExtension degreeCommon base change
2.32.a_a$2$2.1024.a_dau
2.32.q_ey$2$2.1024.a_dau
2.32.i_bg$3$(not in LMFDB)
2.32.ai_bg$6$(not in LMFDB)
2.32.ai_cm$8$(not in LMFDB)
2.32.a_acm$8$(not in LMFDB)
2.32.a_cm$8$(not in LMFDB)
2.32.i_cm$8$(not in LMFDB)
2.32.a_abg$24$(not in LMFDB)
2.32.a_bg$24$(not in LMFDB)