Properties

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

Learn more about

Invariants

Base field:  $\F_{5}$
Dimension:  $2$
L-polynomial:  $1 - 6 x + 17 x^{2} - 30 x^{3} + 25 x^{4}$
Frobenius angles:  $\pm0.0512862249088$, $\pm0.384619558242$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{2}, \sqrt{-3})\)
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
$A(\F_{q^r})$ 7 553 15484 363321 9198847 239754256 6109689607 152926532073 3814701058588 95315867737993

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 0 24 126 580 2940 15342 78204 391492 1953126 9760344

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{2}, \sqrt{-3})\).
Endomorphism algebra over $\overline{\F}_{5}$
The base change of $A$ to $\F_{5^{6}}$ is 1.15625.afm 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-6}) \)$)$
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.a_c$3$2.125.a_afm
2.5.g_r$3$2.125.a_afm
2.5.a_c$6$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.5.a_c$3$2.125.a_afm
2.5.g_r$3$2.125.a_afm
2.5.a_c$6$(not in LMFDB)
2.5.a_ac$12$(not in LMFDB)
2.5.ae_i$24$(not in LMFDB)
2.5.e_i$24$(not in LMFDB)