Properties

Label 2.5.ag_s
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:  $( 1 - 4 x + 5 x^{2} )( 1 - 2 x + 5 x^{2} )$
Frobenius angles:  $\pm0.147583617650$, $\pm0.352416382350$
Angle rank:  $1$ (numerical)
Jacobians:  1

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 1 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 8 640 18056 409600 9746888 244117120 6142546376 153413222400 3822192900488 95367450947200

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 0 26 144 654 3120 15626 78624 392734 1956960 9765626

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
The isogeny class factors as 1.5.ae $\times$ 1.5.ac and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{5}$
The base change of $A$ to $\F_{5^{4}}$ is 1.625.o 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
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.ac_c$2$2.25.a_o
2.5.c_c$2$2.25.a_o
2.5.g_s$2$2.25.a_o
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.5.ac_c$2$2.25.a_o
2.5.c_c$2$2.25.a_o
2.5.g_s$2$2.25.a_o
2.5.ai_ba$4$2.625.bc_cdq
2.5.ae_o$4$2.625.bc_cdq
2.5.a_ag$4$2.625.bc_cdq
2.5.a_g$4$2.625.bc_cdq
2.5.e_o$4$2.625.bc_cdq
2.5.i_ba$4$2.625.bc_cdq
2.5.a_ai$8$(not in LMFDB)
2.5.a_i$8$(not in LMFDB)
2.5.ae_l$12$(not in LMFDB)
2.5.ac_ab$12$(not in LMFDB)
2.5.c_ab$12$(not in LMFDB)
2.5.e_l$12$(not in LMFDB)