Properties

Label 2.5.ae_i
Base Field $\F_{5}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{5}$
Dimension:  $2$
L-polynomial:  $1 - 4 x + 8 x^{2} - 20 x^{3} + 25 x^{4}$
Frobenius angles:  $\pm0.0320471084245$, $\pm0.532047108424$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(i, \sqrt{6})\)
Galois group:  $C_2^2$
Jacobians:  1

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 1 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 10 580 12490 336400 9629050 244129540 6042262810 151912857600 3815627066890 95367440008900

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 2 26 98 534 3082 15626 77338 388894 1953602 9765626

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{6})\).
Endomorphism algebra over $\overline{\F}_{5}$
The base change of $A$ to $\F_{5^{4}}$ is 1.625.abu 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-6}) \)$)$
All geometric endomorphisms are defined over $\F_{5^{4}}$.
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
2.5.e_i$2$2.25.a_abu
2.5.a_ac$8$(not in LMFDB)
2.5.a_c$8$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.5.e_i$2$2.25.a_abu
2.5.a_ac$8$(not in LMFDB)
2.5.a_c$8$(not in LMFDB)
2.5.ag_r$24$(not in LMFDB)
2.5.g_r$24$(not in LMFDB)