Invariants
Base field: | $\F_{5^{2}}$ |
Dimension: | $1$ |
L-polynomial: | $1 + x + 25 x^{2}$ |
Frobenius angles: | $\pm0.531884280429$ |
Angle rank: | $1$ (numerical) |
Number field: | \(\Q(\sqrt{-11}) \) |
Galois group: | $C_2$ |
Jacobians: | $3$ |
Isomorphism classes: | 3 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $27$ | $675$ | $15552$ | $389475$ | $9768627$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $27$ | $675$ | $15552$ | $389475$ | $9768627$ | $244166400$ | $6103414827$ | $152587347075$ | $3814700329152$ | $95367442165875$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{5^{2}}$.
Endomorphism algebra over $\F_{5^{2}}$The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-11}) \). |
Base change
This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{5^{2}}$.
Subfield | Primitive Model |
$\F_{5}$ | 1.5.ad |
$\F_{5}$ | 1.5.d |
Twists
Below is a list of all twists of this isogeny class.
Twist | Extension degree | Common base change |
---|---|---|
1.25.ab | $2$ | 1.625.bx |