Properties

Label 2.625.adp_fbd
Base Field $\F_{5^{4}}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

Learn more about

Invariants

Base field:  $\F_{5^{4}}$
Dimension:  $2$
L-polynomial:  $1 - 93 x + 3409 x^{2} - 58125 x^{3} + 390625 x^{4}$
Frobenius angles:  $\pm0.0831736883744$, $\pm0.147926818132$
Angle rank:  $2$ (numerical)
Number field:  4.0.1573221.2
Galois group:  $D_{4}$

This isogeny class is simple and geometrically simple.

Newton polygon

This isogeny class is ordinary.

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

Point counts

This isogeny class contains a Jacobian, and hence is principally polarizable.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 335817 151874245701 59597901976951809 23283038425816307082213 9094947804637745432907103632 3552713706019809483808652696611509 1387778781349524232410466750244266478913 542101086252238337708396497063652780687826693 211758236813709345984387901597764197868971520469881 82718061255304378336990473610283561270652141291485515776

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 533 388795 244113005 152587720627 95367439891958 59604645232054843 37252902999868383269 23283064365794388672547 14551915228376078167239917 9094947017729459493964414750

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5^{4}}$
The endomorphism algebra of this simple isogeny class is 4.0.1573221.2.
All geometric endomorphisms are defined over $\F_{5^{4}}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.625.dp_fbd$2$(not in LMFDB)