Abelian variety isogeny class 3.7.aj_bp_aew over $\F_{7}$

• Label: 3.7.aj_bp_aew
• Base field: $\F_{7}$
• Dimension: $3$
• $p$-rank: $2$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{7}$
Dimension:	$3$
L-polynomial:	$( 1 - 5 x + 7 x^{2} )( 1 - 4 x + 7 x^{2} )( 1 + 7 x^{2} )$
	$1 - 9 x + 41 x^{2} - 126 x^{3} + 287 x^{4} - 441 x^{5} + 343 x^{6}$
Frobenius angles:	$\pm0.106147807505$, $\pm0.227185525829$, $\pm0.5$
Angle rank:	$1$ (numerical)
Isomorphism classes:	150

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$96$	$119808$	$40569984$	$13681115136$	$4822242966816$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-1$	$51$	$344$	$2375$	$17069$	$118908$	$824291$	$5760239$	$40353608$	$282519771$

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 $\F_{7^{6}}$.

Endomorphism algebra over $\F_{7}$

The isogeny class factors as 1.7.af $\times$ 1.7.ae $\times$ 1.7.a and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.7.af : \(\Q(\sqrt{-3}) \).
• 1.7.ae : \(\Q(\sqrt{-3}) \).
• 1.7.a : \(\Q(\sqrt{-7}) \).

Endomorphism algebra over $\overline{\F}_{7}$

The base change of $A$ to $\F_{7^{6}}$ is 1.117649.la^2 $\times$ 1.117649.bak. The endomorphism algebra for each factor is:

• 1.117649.la^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
• 1.117649.bak : the quaternion algebra over \(\Q\) ramified at $7$ and $\infty$.

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{7^{2}}$
  The base change of $A$ to $\F_{7^{2}}$ is 1.49.al $\times$ 1.49.ac $\times$ 1.49.o. The endomorphism algebra for each factor is:

  • 1.49.al : \(\Q(\sqrt{-3}) \).
  • 1.49.ac : \(\Q(\sqrt{-3}) \).
  • 1.49.o : the quaternion algebra over \(\Q\) ramified at $7$ and $\infty$.
• Endomorphism algebra over $\F_{7^{3}}$
  The base change of $A$ to $\F_{7^{3}}$ is 1.343.au $\times$ 1.343.a $\times$ 1.343.u. The endomorphism algebra for each factor is:

  • 1.343.au : \(\Q(\sqrt{-3}) \).
  • 1.343.a : \(\Q(\sqrt{-7}) \).
  • 1.343.u : \(\Q(\sqrt{-3}) \).

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
3.7.ab_b_ao	$2$	(not in LMFDB)
3.7.b_b_o	$2$	(not in LMFDB)
3.7.ag_ba_adg	$3$	(not in LMFDB)
3.7.ad_r_abq	$3$	(not in LMFDB)
3.7.a_ae_a	$3$	(not in LMFDB)
3.7.a_f_a	$3$	(not in LMFDB)
3.7.a_u_a	$3$	(not in LMFDB)
3.7.d_r_bq	$3$	(not in LMFDB)
3.7.g_ba_dg	$3$	(not in LMFDB)
3.7.j_bp_ew	$3$	(not in LMFDB)