Invariants
Base field: | $\F_{19^{2}}$ |
Dimension: | $1$ |
L-polynomial: | $( 1 - 19 x )^{2}$ |
$1 - 38 x + 361 x^{2}$ | |
Frobenius angles: | $0$, $0$ |
Angle rank: | $0$ (numerical) |
Number field: | \(\Q\) |
Galois group: | Trivial |
Jacobians: | $2$ |
This isogeny class is simple and geometrically simple, 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $324$ | $129600$ | $47032164$ | $16983302400$ | $6131061305604$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $324$ | $129600$ | $47032164$ | $16983302400$ | $6131061305604$ | $2213314824974400$ | $799006683995140644$ | $288441413533654041600$ | $104127350297265866137284$ | $37589973457533696060840000$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{19^{2}}$.
Endomorphism algebra over $\F_{19^{2}}$The endomorphism algebra of this simple isogeny class is the quaternion algebra over \(\Q\) ramified at $19$ and $\infty$. |
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 |
---|---|---|
1.361.bm | $2$ | (not in LMFDB) |
1.361.a | $4$ | (not in LMFDB) |