Properties

Label 2.49.abc_li
Base Field $\F_{7^{2}}$
Dimension $2$
Ordinary No
$p$-rank $0$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

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

This isogeny class is not simple.

Newton polygon

This isogeny class is supersingular.

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

Point counts

This isogeny class contains the Jacobians of 1 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 1296 5308416 13680577296 33177600000000 79773277746356496 191574717809222025216 459984302365352962983696 1104426907919194521600000000 2651730583010213788123328871696 6366805670751667301725999443542016

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

Decomposition and endomorphism algebra

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

Base change

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

SubfieldPrimitive Model
$\F_{7}$2.7.a_ao

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon 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)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon 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)
2.49.ao_du$4$(not in LMFDB)
2.49.a_du$4$(not in LMFDB)
2.49.o_du$4$(not in LMFDB)
2.49.h_bx$5$(not in LMFDB)
2.49.ao_fr$6$(not in LMFDB)
2.49.a_a$8$(not in LMFDB)
2.49.ah_bx$10$(not in LMFDB)
2.49.a_abx$12$(not in LMFDB)