Properties

Label 2.5.ae_o
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:  $( 1 - 2 x + 5 x^{2} )^{2}$
Frobenius angles:  $\pm0.352416382350$, $\pm0.352416382350$
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
$A(\F_{q^r})$ 16 1024 21904 409600 9265936 236913664 6094612624 153413222400 3824074114576 95358193583104

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 2 38 170 654 2962 15158 78010 392734 1957922 9764678

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
The isogeny class factors as 1.5.ac 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
All geometric endomorphisms are defined over $\F_{5}$.

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_g$2$2.25.m_di
2.5.e_o$2$2.25.m_di
2.5.c_ab$3$2.125.bs_bcg
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.5.a_g$2$2.25.m_di
2.5.e_o$2$2.25.m_di
2.5.c_ab$3$2.125.bs_bcg
2.5.ai_ba$4$2.625.bc_cdq
2.5.ag_s$4$2.625.bc_cdq
2.5.ac_c$4$2.625.bc_cdq
2.5.a_ag$4$2.625.bc_cdq
2.5.c_c$4$2.625.bc_cdq
2.5.g_s$4$2.625.bc_cdq
2.5.i_ba$4$2.625.bc_cdq
2.5.ac_ab$6$(not in LMFDB)
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.e_l$12$(not in LMFDB)