Properties

Label 2.5.ai_ba
Base Field $\F_{5}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian No

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Invariants

Base field:  $\F_{5}$
Dimension:  $2$
L-polynomial:  $( 1 - 4 x + 5 x^{2} )^{2}$
Frobenius angles:  $\pm0.147583617650$, $\pm0.147583617650$
Angle rank:  $1$ (numerical)
Jacobians:  0

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 is principally polarizable, but does not contain a Jacobian.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 4 400 14884 409600 10252804 251539600 6190857124 153413222400 3820312611844 95376709210000

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -2 14 118 654 3278 16094 79238 392734 1955998 9766574

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
The isogeny class factors as 1.5.ae 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_ag$2$2.25.am_di
2.5.i_ba$2$2.25.am_di
2.5.e_l$3$2.125.ai_kg
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.5.a_ag$2$2.25.am_di
2.5.i_ba$2$2.25.am_di
2.5.e_l$3$2.125.ai_kg
2.5.ag_s$4$2.625.bc_cdq
2.5.ae_o$4$2.625.bc_cdq
2.5.ac_c$4$2.625.bc_cdq
2.5.a_g$4$2.625.bc_cdq
2.5.c_c$4$2.625.bc_cdq
2.5.e_o$4$2.625.bc_cdq
2.5.g_s$4$2.625.bc_cdq
2.5.ae_l$6$(not in LMFDB)
2.5.a_ai$8$(not in LMFDB)
2.5.a_i$8$(not in LMFDB)
2.5.ac_ab$12$(not in LMFDB)
2.5.c_ab$12$(not in LMFDB)