Properties

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

Learn more about

Invariants

Base field:  $\F_{5}$
Dimension:  $2$
L-polynomial:  $1 - 2 x - x^{2} - 10 x^{3} + 25 x^{4}$
Frobenius angles:  $\pm0.0190830490162$, $\pm0.685749715683$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\zeta_{12})\)
Galois group:  $C_2^2$
Jacobians:  0

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 is not principally polarizable, and therefore does not contain a Jacobian.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 13 481 10816 381433 9512893 236913664 6098909557 152176891113 3805339729984 95372051006401

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 4 20 82 612 3044 15158 78068 389572 1948330 9766100

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{12})\).
Endomorphism algebra over $\overline{\F}_{5}$
The base change of $A$ to $\F_{5^{3}}$ is 1.125.aw 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
All geometric endomorphisms are defined over $\F_{5^{3}}$.

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.c_ab$2$2.25.ag_l
2.5.e_o$3$2.125.abs_bcg
2.5.ae_l$4$2.625.ao_aqn
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.5.c_ab$2$2.25.ag_l
2.5.e_o$3$2.125.abs_bcg
2.5.ae_l$4$2.625.ao_aqn
2.5.e_l$4$2.625.ao_aqn
2.5.ae_o$6$(not in LMFDB)
2.5.a_g$6$(not in LMFDB)
2.5.ai_ba$12$(not in LMFDB)
2.5.ag_s$12$(not in LMFDB)
2.5.ac_c$12$(not in LMFDB)
2.5.a_ag$12$(not in LMFDB)
2.5.c_c$12$(not in LMFDB)
2.5.g_s$12$(not in LMFDB)
2.5.i_ba$12$(not in LMFDB)
2.5.a_ai$24$(not in LMFDB)
2.5.a_i$24$(not in LMFDB)