Invariants
Base field: | $\F_{5^{4}}$ |
Dimension: | $1$ |
L-polynomial: | $1 - 31 x + 625 x^{2}$ |
Frobenius angles: | $\pm0.287132586257$ |
Angle rank: | $1$ (numerical) |
Number field: | \(\Q(\sqrt{-19}) \) |
Galois group: | $C_2$ |
This isogeny class is simple and geometrically simple, not 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $595$ | $390915$ | $244168960$ | $152588588355$ | $95367435561475$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $595$ | $390915$ | $244168960$ | $152588588355$ | $95367435561475$ | $59604644460856320$ | $37252902972418046515$ | $23283064365205312914435$ | $14551915228368846341198080$ | $9094947017729457740956791075$ |
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 \(\Q(\sqrt{-19}) \). |
Base change
This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{5^{4}}$.
Subfield | Primitive Model |
$\F_{5}$ | 1.5.ab |
$\F_{5}$ | 1.5.b |
Twists
Below is a list of all twists of this isogeny class.
Twist | Extension degree | Common base change |
---|---|---|
1.625.bf | $2$ | (not in LMFDB) |