Properties

Label 2.7.ae_i
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:  $1 - 4 x + 8 x^{2} - 28 x^{3} + 49 x^{4}$
Frobenius angles:  $\pm0.0704914820143$, $\pm0.570491482014$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(i, \sqrt{10})\)
Galois group:  $C_2^2$
Jacobians:  3

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 3 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 26 2340 101114 5475600 284431706 13841495460 676699584314 33243988377600 1629091518163226 79792266455518500

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 4 50 292 2278 16924 117650 821692 5766718 40370404 282475250

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7}$
The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{10})\).
Endomorphism algebra over $\overline{\F}_{7}$
The base change of $A$ to $\F_{7^{4}}$ is 1.2401.ack 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-10}) \)$)$
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 are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.7.e_i$2$2.49.a_ack
2.7.e_i$4$(not in LMFDB)
2.7.a_ag$8$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.7.e_i$2$2.49.a_ack
2.7.e_i$4$(not in LMFDB)
2.7.a_ag$8$(not in LMFDB)
2.7.a_g$8$(not in LMFDB)