Invariants
Base field: | $\F_{5^{4}}$ |
Dimension: | $1$ |
L-polynomial: | $( 1 - 25 x )^{2}$ |
$1 - 50 x + 625 x^{2}$ | |
Frobenius angles: | $0$, $0$ |
Angle rank: | $0$ (numerical) |
Number field: | \(\Q\) |
Galois group: | Trivial |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $576$ | $389376$ | $244109376$ | $152587109376$ | $95367412109376$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $576$ | $389376$ | $244109376$ | $152587109376$ | $95367412109376$ | $59604644287109376$ | $37252902972412109376$ | $23283064365081787109376$ | $14551915228359222412109376$ | $9094947017729091644287109376$ |
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 the quaternion algebra over \(\Q\) ramified at $5$ and $\infty$. |
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.a |
Twists
Below is a list of all twists of this isogeny class.
Twist | Extension degree | Common base change |
---|---|---|
1.625.by | $2$ | (not in LMFDB) |
1.625.z | $3$ | (not in LMFDB) |
1.625.az | $6$ | (not in LMFDB) |