Properties

Label 2.16.ah_bg
Base field $\F_{2^{4}}$
Dimension $2$
$p$-rank $1$
Ordinary no
Supersingular no
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{2^{4}}$
Dimension:  $2$
L-polynomial:  $( 1 - 7 x + 16 x^{2} )( 1 + 16 x^{2} )$
  $1 - 7 x + 32 x^{2} - 112 x^{3} + 256 x^{4}$
Frobenius angles:  $\pm0.160861246510$, $\pm0.5$
Angle rank:  $1$ (numerical)
Jacobians:  $4$

This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

$p$-rank:  $1$
Slopes:  $[0, 1/2, 1/2, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $170$ $69360$ $16756730$ $4276044000$ $1101267994250$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $10$ $272$ $4090$ $65248$ $1050250$ $16793552$ $268465690$ $4294917568$ $68719562410$ $1099513023152$

Jacobians and polarizations

This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{2^{4}}$
The isogeny class factors as 1.16.ah $\times$ 1.16.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 $\overline{\F}_{2^{4}}$
The base change of $A$ to $\F_{2^{8}}$ is 1.256.ar $\times$ 1.256.bg. The endomorphism algebra for each factor is:

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.h_bg$2$2.256.p_abg
2.16.ap_dk$4$(not in LMFDB)
2.16.ab_ay$4$(not in LMFDB)

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

TwistExtension degreeCommon base change
2.16.h_bg$2$2.256.p_abg
2.16.ap_dk$4$(not in LMFDB)
2.16.ab_ay$4$(not in LMFDB)
2.16.b_ay$4$(not in LMFDB)
2.16.p_dk$4$(not in LMFDB)
2.16.al_ci$12$(not in LMFDB)
2.16.ad_e$12$(not in LMFDB)
2.16.d_e$12$(not in LMFDB)
2.16.l_ci$12$(not in LMFDB)