Properties

Label 2.7.ag_s
Base Field $\F_{7}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{7}$
Dimension:  $2$
L-polynomial:  $1 - 6 x + 18 x^{2} - 42 x^{3} + 49 x^{4}$
Frobenius angles:  $\pm0.0461154155528$, $\pm0.453884584447$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(i, \sqrt{5})\)
Galois group:  $C_2^2$
Jacobians:  2

This isogeny class is simple but not geometrically simple.

Newton polygon

This isogeny class is ordinary.

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

Point counts

This isogeny class contains the Jacobians of 2 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 20 2320 111620 5382400 276390500 13841326480 678702544820 33210785894400 1627783961277620 79792266845818000

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 2 50 326 2238 16442 117650 824126 5760958 40338002 282475250

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7}$
The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{5})\).
Endomorphism algebra over $\overline{\F}_{7}$
The base change of $A$ to $\F_{7^{4}}$ is 1.2401.ade 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-5}) \)$)$
All geometric endomorphisms are defined over $\F_{7^{4}}$.
Remainder of endomorphism lattice by field

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.7.g_s$2$2.49.a_ade
2.7.a_ae$8$(not in LMFDB)
2.7.a_e$8$(not in LMFDB)