Properties

Label 3.4.ag_r_abk
Base field $\F_{2^{2}}$
Dimension $3$
$p$-rank $2$
Ordinary no
Supersingular no
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian no

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Invariants

Base field:  $\F_{2^{2}}$
Dimension:  $3$
L-polynomial:  $( 1 - 2 x )^{2}( 1 - 3 x + 4 x^{2} )( 1 + x + 4 x^{2} )$
  $1 - 6 x + 17 x^{2} - 36 x^{3} + 68 x^{4} - 96 x^{5} + 64 x^{6}$
Frobenius angles:  $0$, $0$, $\pm0.230053456163$, $\pm0.580430623255$
Angle rank:  $2$ (numerical)

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $12$ $3456$ $195804$ $15552000$ $1129224972$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-1$ $15$ $47$ $239$ $1079$ $4023$ $15791$ $64799$ $260903$ $1043655$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{2^{2}}$
The isogeny class factors as 1.4.ae $\times$ 1.4.ad $\times$ 1.4.b and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. 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
3.4.ai_bf_acy$2$3.16.ac_ah_ai
3.4.ac_b_ae$2$3.16.ac_ah_ai
3.4.a_ab_am$2$3.16.ac_ah_ai
3.4.a_ab_m$2$3.16.ac_ah_ai
3.4.c_b_e$2$3.16.ac_ah_ai
3.4.g_r_bk$2$3.16.ac_ah_ai
3.4.i_bf_cy$2$3.16.ac_ah_ai
3.4.a_f_ag$3$(not in LMFDB)

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

TwistExtension degreeCommon base change
3.4.ai_bf_acy$2$3.16.ac_ah_ai
3.4.ac_b_ae$2$3.16.ac_ah_ai
3.4.a_ab_am$2$3.16.ac_ah_ai
3.4.a_ab_m$2$3.16.ac_ah_ai
3.4.c_b_e$2$3.16.ac_ah_ai
3.4.g_r_bk$2$3.16.ac_ah_ai
3.4.i_bf_cy$2$3.16.ac_ah_ai
3.4.a_f_ag$3$(not in LMFDB)
3.4.ae_p_abg$4$(not in LMFDB)
3.4.ac_j_aq$4$(not in LMFDB)
3.4.c_j_q$4$(not in LMFDB)
3.4.e_p_bg$4$(not in LMFDB)
3.4.ag_x_acc$6$(not in LMFDB)
3.4.ae_n_aba$6$(not in LMFDB)
3.4.ac_h_ak$6$(not in LMFDB)
3.4.a_f_g$6$(not in LMFDB)
3.4.c_h_k$6$(not in LMFDB)
3.4.e_n_ba$6$(not in LMFDB)
3.4.g_x_cc$6$(not in LMFDB)