Invariants
Base field: | $\F_{13^{2}}$ |
Dimension: | $1$ |
L-polynomial: | $1 + 17 x + 169 x^{2}$ |
Frobenius angles: | $\pm0.726845650153$ |
Angle rank: | $1$ (numerical) |
Number field: | \(\Q(\sqrt{-43}) \) |
Galois group: | $C_2$ |
Jacobians: | $5$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $187$ | $28611$ | $4823104$ | $815785443$ | $137858187907$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $187$ | $28611$ | $4823104$ | $815785443$ | $137858187907$ | $23298081041664$ | $3937376506439563$ | $665416607820253443$ | $112455406954722035776$ | $19004963775064135075251$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 5 curves (of which all are hyperelliptic), and hence is principally polarizable:
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{13^{2}}$.
Endomorphism algebra over $\F_{13^{2}}$The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-43}) \). |
Base change
This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{13^{2}}$.
Subfield | Primitive Model |
$\F_{13}$ | 1.13.ad |
$\F_{13}$ | 1.13.d |
Twists
Below is a list of all twists of this isogeny class.
Twist | Extension degree | Common base change |
---|---|---|
1.169.ar | $2$ | (not in LMFDB) |