Invariants
Base field: | $\F_{31}$ |
Dimension: | $1$ |
L-polynomial: | $1 + 3 x + 31 x^{2}$ |
Frobenius angles: | $\pm0.586827998080$ |
Angle rank: | $1$ (numerical) |
Number field: | \(\Q(\sqrt{-115}) \) |
Galois group: | $C_2$ |
Jacobians: | $2$ |
Isomorphism classes: | 2 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $35$ | $1015$ | $29540$ | $922635$ | $28639625$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $35$ | $1015$ | $29540$ | $922635$ | $28639625$ | $887499760$ | $27512301215$ | $852892097715$ | $26439628679660$ | $819628234555375$ |
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 $\F_{31}$.
Endomorphism algebra over $\F_{31}$The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-115}) \). |
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 |
---|---|---|
1.31.ad | $2$ | (not in LMFDB) |