Abelian variety isogeny class 5.3.g_ba_da_hh_nw over $\F_{3}$

• Label: 5.3.g_ba_da_hh_nw
• Base field: $\F_{3}$
• Dimension: $5$
• $p$-rank: $2$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes

Invariants

Base field:	$\F_{3}$
Dimension:	$5$
L-polynomial:	$( 1 + x + 3 x^{2} )( 1 + 2 x + 3 x^{2} )( 1 + 3 x + 3 x^{2} )( 1 + 3 x^{2} )^{2}$
	$1 + 6 x + 26 x^{2} + 78 x^{3} + 189 x^{4} + 360 x^{5} + 567 x^{6} + 702 x^{7} + 702 x^{8} + 486 x^{9} + 243 x^{10}$
Frobenius angles:	$\pm0.5$, $\pm0.5$, $\pm0.593214749339$, $\pm0.695913276015$, $\pm0.833333333333$
Angle rank:	$2$ (numerical)
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3360$	$322560$	$7902720$	$2683699200$	$873991456800$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$10$	$26$	$10$	$62$	$250$	$836$	$2110$	$6398$	$19630$	$59786$

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 $\F_{3^{6}}$.

Endomorphism algebra over $\F_{3}$

The isogeny class factors as 1.3.a^2 $\times$ 1.3.b $\times$ 1.3.c $\times$ 1.3.d and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.3.a^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
• 1.3.b : \(\Q(\sqrt{-11}) \).
• 1.3.c : \(\Q(\sqrt{-2}) \).
• 1.3.d : \(\Q(\sqrt{-3}) \).

Endomorphism algebra over $\overline{\F}_{3}$

The base change of $A$ to $\F_{3^{6}}$ is 1.729.abu $\times$ 1.729.ak $\times$ 1.729.cc^3. The endomorphism algebra for each factor is:

• 1.729.abu : \(\Q(\sqrt{-2}) \).
• 1.729.ak : \(\Q(\sqrt{-11}) \).
• 1.729.cc^3 : $\mathrm{M}_{3}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{3^{2}}$
  The base change of $A$ to $\F_{3^{2}}$ is 1.9.ad $\times$ 1.9.c $\times$ 1.9.f $\times$ 1.9.g^2. The endomorphism algebra for each factor is:

  • 1.9.ad : \(\Q(\sqrt{-3}) \).
  • 1.9.c : \(\Q(\sqrt{-2}) \).
  • 1.9.f : \(\Q(\sqrt{-11}) \).
  • 1.9.g^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
• Endomorphism algebra over $\F_{3^{3}}$
  The base change of $A$ to $\F_{3^{3}}$ is 1.27.ak $\times$ 1.27.ai $\times$ 1.27.a^3. The endomorphism algebra for each factor is:

  • 1.27.ak : \(\Q(\sqrt{-2}) \).
  • 1.27.ai : \(\Q(\sqrt{-11}) \).
  • 1.27.a^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
5.3.ag_ba_ada_hh_anw	$2$	(not in LMFDB)
5.3.ae_q_abq_dv_agy	$2$	(not in LMFDB)
5.3.ac_k_as_bt_acu	$2$	(not in LMFDB)
5.3.a_i_ag_bb_abk	$2$	(not in LMFDB)
5.3.a_i_g_bb_bk	$2$	(not in LMFDB)
5.3.c_k_s_bt_cu	$2$	(not in LMFDB)
5.3.e_q_bq_dv_gy	$2$	(not in LMFDB)
5.3.ag_r_abk_dd_agg	$3$	(not in LMFDB)
5.3.ad_i_av_bt_acu	$3$	(not in LMFDB)
5.3.a_ab_ag_j_s	$3$	(not in LMFDB)
5.3.a_i_ag_bb_abk	$3$	(not in LMFDB)
5.3.d_i_j_j_a	$3$	(not in LMFDB)
5.3.d_r_bk_ee_gg	$3$	(not in LMFDB)
5.3.g_r_y_j_as	$3$	(not in LMFDB)
5.3.j_bs_fr_of_bbs	$3$	(not in LMFDB)
5.3.m_ct_kk_bcb_cec	$3$	(not in LMFDB)