Invariants
Base field: | $\F_{499}$ |
Dimension: | $1$ |
L-polynomial: | $1 - 15 x + 499 x^{2}$ |
Frobenius angles: | $\pm0.391011583697$ |
Angle rank: | $1$ (numerical) |
Number field: | \(\Q(\sqrt{-1771}) \) |
Galois group: | $C_2$ |
Jacobians: | $8$ |
This isogeny class is simple and geometrically simple, 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $485$ | $249775$ | $124270580$ | $62001398475$ | $30938736488675$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $485$ | $249775$ | $124270580$ | $62001398475$ | $30938736488675$ | $15438434888203600$ | $7703779070632501145$ | $3844185754482104368275$ | $1918248691429469340513740$ | $957206097023328683165119375$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{499}$.
Endomorphism algebra over $\F_{499}$The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-1771}) \). |
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 |
---|---|---|
1.499.p | $2$ | (not in LMFDB) |