Properties

Label 2.16.aq_ds
Base Field $\F_{2^{4}}$
Dimension $2$
Ordinary No
$p$-rank $0$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{2^{4}}$
Dimension:  $2$
L-polynomial:  $( 1 - 4 x )^{4}$
Frobenius angles:  $0$, $0$, $0$, $0$
Angle rank:  $0$ (numerical)
Jacobians:  1

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 contains the Jacobians of 1 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 81 50625 15752961 4228250625 1095222947841 281200199450625 72040003462430721 18445618199572250625 4722294425687923097601 1208921207935207812890625

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 1 193 3841 64513 1044481 16760833 268369921 4294705153 68718428161 1099507433473

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
The isogeny class factors as 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 isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{4}}$.

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

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.16.a_abg$2$2.256.acm_chc
2.16.q_ds$2$2.256.acm_chc
2.16.ae_a$3$(not in LMFDB)
2.16.i_bw$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.16.a_abg$2$2.256.acm_chc
2.16.q_ds$2$2.256.acm_chc
2.16.ae_a$3$(not in LMFDB)
2.16.i_bw$3$(not in LMFDB)
2.16.ai_bg$4$(not in LMFDB)
2.16.a_bg$4$(not in LMFDB)
2.16.i_bg$4$(not in LMFDB)
2.16.e_q$5$(not in LMFDB)
2.16.am_cm$6$(not in LMFDB)
2.16.ai_bw$6$(not in LMFDB)
2.16.a_q$6$(not in LMFDB)
2.16.e_a$6$(not in LMFDB)
2.16.m_cm$6$(not in LMFDB)
2.16.a_a$8$(not in LMFDB)
2.16.ae_q$10$(not in LMFDB)
2.16.ae_bg$12$(not in LMFDB)
2.16.a_aq$12$(not in LMFDB)
2.16.e_bg$12$(not in LMFDB)