Properties

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

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Invariants

Base field:  $\F_{2^{4}}$
Dimension:  $2$
L-polynomial:  $( 1 - 4 x )^{2}( 1 - 4 x + 16 x^{2} )$
Frobenius angles:  $0$, $0$, $\pm0.333333333333$
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
$A(\F_{q^r})$ 117 61425 16769025 4278189825 1096294593537 281200199450625 72044400972087297 18446462598732775425 4722366482732206260225 1208924666693124566810625

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 5 241 4097 65281 1045505 16760833 268386305 4294901761 68719476737 1099510579201

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
The isogeny class factors as 1.16.ai $\times$ 1.16.ae 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 $\overline{\F}_{2^{4}}$
The base change of $A$ to $\F_{2^{24}}$ is 1.16777216.amdc 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 $\F_{2^{24}}$.
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
2.16.ae_a$2$2.256.aq_a
2.16.e_a$2$2.256.aq_a
2.16.m_cm$2$2.256.aq_a
2.16.a_abg$3$(not in LMFDB)
2.16.a_q$3$(not in LMFDB)
2.16.m_cm$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.16.ae_a$2$2.256.aq_a
2.16.e_a$2$2.256.aq_a
2.16.m_cm$2$2.256.aq_a
2.16.a_abg$3$(not in LMFDB)
2.16.a_q$3$(not in LMFDB)
2.16.m_cm$3$(not in LMFDB)
2.16.ae_bg$4$(not in LMFDB)
2.16.e_bg$4$(not in LMFDB)
2.16.aq_ds$6$(not in LMFDB)
2.16.ai_bw$6$(not in LMFDB)
2.16.i_bw$6$(not in LMFDB)
2.16.q_ds$6$(not in LMFDB)
2.16.ai_bg$12$(not in LMFDB)
2.16.a_aq$12$(not in LMFDB)
2.16.a_bg$12$(not in LMFDB)
2.16.i_bg$12$(not in LMFDB)
2.16.a_a$24$(not in LMFDB)
2.16.ae_q$30$(not in LMFDB)
2.16.e_q$30$(not in LMFDB)