Abelian variety isogeny class 2.49.abc_li over $\F_{7^{2}}$

• Label: 2.49.abc_li
• Base field: $\F_{7^{2}}$
• Dimension: $2$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: no
• Geometrically simple: no
• Primitive: no
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{7^{2}}$
Dimension:	$2$
L-polynomial:	$( 1 - 7 x )^{4}$
	$1 - 28 x + 294 x^{2} - 1372 x^{3} + 2401 x^{4}$
Frobenius angles:	$0$, $0$, $0$, $0$
Angle rank:	$0$ (numerical)
Jacobians:	$1$

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

Newton polygon

This isogeny class is supersingular.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1296$	$5308416$	$13680577296$	$33177600000000$	$79773277746356496$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$22$	$2206$	$116278$	$5755198$	$282408022$	$13840816606$	$678219778678$	$33232907510398$	$1628413436496022$	$79792265167711006$

Jacobians and polarizations

This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

• $y^2=ax^5+6ax$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{7^{2}}$.

Endomorphism algebra over $\F_{7^{2}}$

The isogeny class factors as 1.49.ao^2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $7$ and $\infty$.

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{7^{2}}$.

Subfield	Primitive Model
$\F_{7}$	2.7.a_ao

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.49.a_adu	$2$	(not in LMFDB)
2.49.bc_li	$2$	(not in LMFDB)
2.49.o_fr	$3$	(not in LMFDB)