Abelian variety isogeny class 2.7.ad_k over $\F_{7}$

• Label: 2.7.ad_k
• Base field: $\F_{7}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{7}$
Dimension:	$2$
L-polynomial:	$( 1 - 4 x + 7 x^{2} )( 1 + x + 7 x^{2} )$
	$1 - 3 x + 10 x^{2} - 21 x^{3} + 49 x^{4}$
Frobenius angles:	$\pm0.227185525829$, $\pm0.560518859162$
Angle rank:	$1$ (numerical)
Jacobians:	$6$
Isomorphism classes:	28
Cyclic group of points:	no
Non-cyclic primes:	$2, 3$

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

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$36$	$3024$	$117936$	$5818176$	$290071836$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$5$	$61$	$344$	$2425$	$17255$	$118222$	$821273$	$5760529$	$40353608$	$282442261$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):

• $y^2=6 x^6+6 x^5+2 x^4+3 x^3+x^2+4 x+6$
• $y^2=4 x^6+2 x^5+4 x^4+3 x^3+3 x^2+2 x+1$
• $y^2=5 x^5+3 x^4+x^3+4 x^2+2 x+6$
• $y^2=3 x^6+2 x^5+6 x^4+6 x^3+x^2+3 x+3$
• $y^2=x^6+x^3+5$
• $y^2=3 x^6+5 x^5+3 x^4+5 x^3+6 x^2+4 x+5$

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.ae $\times$ 1.7.b 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.ae : \(\Q(\sqrt{-3}) \).
• 1.7.b : \(\Q(\sqrt{-3}) \).

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

The base change of $A$ to $\F_{7^{6}}$ is 1.117649.la^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

Remainder of endomorphism lattice by field

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

  • 1.49.ac : \(\Q(\sqrt{-3}) \).
  • 1.49.n : \(\Q(\sqrt{-3}) \).
• Endomorphism algebra over $\F_{7^{3}}$
  The base change of $A$ to $\F_{7^{3}}$ is 1.343.au $\times$ 1.343.u. The endomorphism algebra for each factor is:

  • 1.343.au : \(\Q(\sqrt{-3}) \).
  • 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
2.7.af_s	$2$	2.49.l_cu
2.7.d_k	$2$	2.49.l_cu
2.7.f_s	$2$	2.49.l_cu
2.7.aj_bi	$3$	2.343.a_la
2.7.ag_t	$3$	2.343.a_la
2.7.a_al	$3$	2.343.a_la
2.7.a_ac	$3$	2.343.a_la
2.7.a_n	$3$	2.343.a_la
2.7.d_k	$3$	2.343.a_la
2.7.g_t	$3$	2.343.a_la
2.7.j_bi	$3$	2.343.a_la