Properties

Label 2.49.ay_ji
Base Field $\F_{7^{2}}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{7^{2}}$
Dimension:  $2$
L-polynomial:  $( 1 - 12 x + 49 x^{2} )^{2}$
Frobenius angles:  $\pm0.172237328522$, $\pm0.172237328522$
Angle rank:  $1$ (numerical)
Jacobians:  5

This isogeny class is not 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 5 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 1444 5550736 13849994596 33263917830144 79809480702695524 191587709240781609616 459988320399300514995364 1104427961044595342275313664 2651730804630061440313388028196 6366805702924223586474522587602576

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 26 2310 117722 5770174 282536186 13841755206 678225703034 33232939199614 1628413572591578 79792265570914950

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
The isogeny class factors as 1.49.am 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-13}) \)$)$
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_am

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.49.a_abu$2$(not in LMFDB)
2.49.y_ji$2$(not in LMFDB)
2.49.m_dr$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.49.a_abu$2$(not in LMFDB)
2.49.y_ji$2$(not in LMFDB)
2.49.m_dr$3$(not in LMFDB)
2.49.a_bu$4$(not in LMFDB)
2.49.am_dr$6$(not in LMFDB)