Properties

Label 3.8.l_ci_ia
Base field $\F_{2^{3}}$
Dimension $3$
$p$-rank $1$
Ordinary no
Supersingular no
Simple no
Geometrically simple no
Primitive no
Principally polarizable yes

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Invariants

Base field:  $\F_{2^{3}}$
Dimension:  $3$
L-polynomial:  $( 1 + 4 x + 8 x^{2} )( 1 + 7 x + 24 x^{2} + 56 x^{3} + 64 x^{4} )$
  $1 + 11 x + 60 x^{2} + 208 x^{3} + 480 x^{4} + 704 x^{5} + 512 x^{6}$
Frobenius angles:  $\pm0.581839774401$, $\pm0.750000000000$, $\pm0.941488805765$
Angle rank:  $2$ (numerical)

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $1976$ $256880$ $127726664$ $68157970400$ $35519271367736$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $20$ $64$ $488$ $4064$ $33080$ $261712$ $2097192$ $16772032$ $134249240$ $1073703984$

Jacobians and polarizations

This isogeny class is principally polarizable and contains no Jacobian of a hyperelliptic curve, but it is unknown whether it contains a Jacobian of a non-hyperelliptic curve.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{12}}$.

Endomorphism algebra over $\F_{2^{3}}$
The isogeny class factors as 1.8.e $\times$ 2.8.h_y 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 $\overline{\F}_{2^{3}}$
The base change of $A$ to $\F_{2^{12}}$ is 1.4096.ey $\times$ 2.4096.agf_vki. The endomorphism algebra for each factor is:
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 $\F_{2^{3}}$.

SubfieldPrimitive Model
$\F_{2}$3.2.ab_a_e

Twists

Below are some of the twists of this isogeny class.

TwistExtension degreeCommon base change
3.8.al_ci_aia$2$(not in LMFDB)
3.8.ad_e_aq$2$(not in LMFDB)
3.8.d_e_q$2$(not in LMFDB)

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

TwistExtension degreeCommon base change
3.8.al_ci_aia$2$(not in LMFDB)
3.8.ad_e_aq$2$(not in LMFDB)
3.8.d_e_q$2$(not in LMFDB)
3.8.ah_bg_aei$8$(not in LMFDB)
3.8.h_bg_ei$8$(not in LMFDB)