Invariants
Base field: | $\F_{13^{2}}$ |
Dimension: | $1$ |
L-polynomial: | $( 1 + 13 x )^{2}$ |
$1 + 26 x + 169 x^{2}$ | |
Frobenius angles: | $1$, $1$ |
Angle rank: | $0$ (numerical) |
Number field: | \(\Q\) |
Galois group: | Trivial |
Jacobians: | $1$ |
This isogeny class is simple and geometrically simple, not primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is supersingular.
$p$-rank: | $0$ |
Slopes: | $[1/2, 1/2]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $196$ | $28224$ | $4831204$ | $815673600$ | $137859234436$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $196$ | $28224$ | $4831204$ | $815673600$ | $137859234436$ | $23298075468864$ | $3937376511196324$ | $665416607551718400$ | $112455406973166391876$ | $19004963774605082455104$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is 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 the quaternion algebra over \(\Q\) ramified at $13$ and $\infty$. |
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.a |
Twists
Below is a list of all twists of this isogeny class.
Twist | Extension degree | Common base change |
---|---|---|
1.169.aba | $2$ | (not in LMFDB) |