Properties

Label 2.5.ad_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:  $1 - 3 x + 8 x^{2} - 15 x^{3} + 25 x^{4}$
Frobenius angles:  $\pm0.206741677780$, $\pm0.540075011113$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-3}, \sqrt{17})\)
Galois group:  $C_2^2$
Jacobians:  4

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 4 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 16 832 15808 389376 10332496 249892864 6077147344 152101168128 3814694762944 95326492186432

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 3 33 126 625 3303 15990 77787 389377 1953126 9761433

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{17})\).
Endomorphism algebra over $\overline{\F}_{5}$
The base change of $A$ to $\F_{5^{6}}$ is 1.15625.ha 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-51}) \)$)$
All geometric endomorphisms are defined over $\F_{5^{6}}$.
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.d_i$2$2.25.h_y
2.5.a_ah$3$2.125.a_ha
2.5.d_i$3$2.125.a_ha
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.5.d_i$2$2.25.h_y
2.5.a_ah$3$2.125.a_ha
2.5.d_i$3$2.125.a_ha
2.5.a_h$12$(not in LMFDB)