Properties

Label 2.5.ag_r
Base field $\F_{5}$
Dimension $2$
$p$-rank $2$
Ordinary yes
Supersingular no
Simple yes
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

Related objects

Downloads

Learn more

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:  $1$
Isomorphism classes:  1

This isogeny class is simple but not geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $2$
Slopes:  $[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $7$ $553$ $15484$ $363321$ $9198847$

Point counts of the curve

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

Jacobians and polarizations

This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{5^{6}}$.

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 $\mathrm{M}_{2}($\(\Q(\sqrt{-6}) \)$)$
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)