Properties

Label 3.2.ad_d_ac
Base field $\F_{2}$
Dimension $3$
$p$-rank $2$
Ordinary no
Supersingular no
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

Learn more

Invariants

Base field:  $\F_{2}$
Dimension:  $3$
L-polynomial:  $( 1 - 2 x + 2 x^{2} )( 1 - x - x^{2} - 2 x^{3} + 4 x^{4} )$
  $1 - 3x + 3x^{2} - 2x^{3} + 6x^{4} - 12x^{5} + 8x^{6}$
Frobenius angles:  $\pm0.0516399385854$, $\pm0.250000000000$, $\pm0.718306605252$
Angle rank:  $1$ (numerical)
Jacobians:  1

This isogeny class is not simple.

Newton polygon

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

Point counts

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

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $1$ $35$ $208$ $6475$ $30791$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $0$ $2$ $3$ $26$ $30$ $47$ $126$ $194$ $471$ $1082$

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
The isogeny class factors as 1.2.ac $\times$ 2.2.ab_ab 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 $\overline{\F}_{2}$
The base change of $A$ to $\F_{2^{12}}$ is 1.4096.bv 2 $\times$ 1.4096.ey. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{2^{12}}$.
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
3.2.ab_ab_g$2$3.4.ad_j_ay
3.2.b_ab_ag$2$3.4.ad_j_ay
3.2.d_d_c$2$3.4.ad_j_ay
3.2.a_d_ac$3$3.8.ag_j_e
Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
3.2.ab_ab_g$2$3.4.ad_j_ay
3.2.b_ab_ag$2$3.4.ad_j_ay
3.2.d_d_c$2$3.4.ad_j_ay
3.2.a_d_ac$3$3.8.ag_j_e
3.2.ae_l_as$6$(not in LMFDB)
3.2.ac_f_ag$6$(not in LMFDB)
3.2.a_d_c$6$(not in LMFDB)
3.2.c_f_g$6$(not in LMFDB)
3.2.e_l_s$6$(not in LMFDB)
3.2.ab_b_ae$8$(not in LMFDB)
3.2.b_b_e$8$(not in LMFDB)
3.2.ac_ab_g$12$(not in LMFDB)
3.2.c_ab_ag$12$(not in LMFDB)
3.2.e_l_s$12$(not in LMFDB)
3.2.ac_h_ai$24$(not in LMFDB)
3.2.a_ab_a$24$(not in LMFDB)
3.2.a_f_a$24$(not in LMFDB)
3.2.c_h_i$24$(not in LMFDB)

Additional information

This is the isogeny class of the Jacobian of a function field of class number 1. It also appears as a sporadic example in the classification of abelian varieties with one rational point.