Invariants
Base field: | $\F_{5^{4}}$ |
Dimension: | $1$ |
L-polynomial: | $1 - 12 x + 625 x^{2}$ |
Frobenius angles: | $\pm0.422852553541$ |
Angle rank: | $1$ (numerical) |
Number field: | \(\Q(\sqrt{-589}) \) |
Galois group: | $C_2$ |
This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
$p$-rank: | $1$ |
Slopes: | $[0, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $614$ | $391732$ | $244161398$ | $152587448640$ | $95367413354294$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $614$ | $391732$ | $244161398$ | $152587448640$ | $95367413354294$ | $59604644832195892$ | $37252902996729761318$ | $23283064365496787047680$ | $14551915228360600558592774$ | $9094947017729138724075657652$ |
Jacobians and polarizations
This isogeny class contains a Jacobian, and hence is principally polarizable.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{5^{4}}$.
Endomorphism algebra over $\F_{5^{4}}$The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-589}) \). |
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
Twist | Extension degree | Common base change |
---|---|---|
1.625.m | $2$ | (not in LMFDB) |