Invariants
| Base field: | $\F_{3^{5}}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 - 14 x + 243 x^{2}$ |
| Frobenius angles: | $\pm0.351762429531$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-194}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $20$ |
| Isomorphism classes: | 20 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $230$ | $59340$ | $14356370$ | $3486818400$ | $847287272150$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $230$ | $59340$ | $14356370$ | $3486818400$ | $847287272150$ | $205891105111020$ | $50031545046191330$ | $12157665464874633600$ | $2954312706645114001670$ | $717897987691758810746700$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 20 curves (of which 0 are hyperelliptic):
- $y^2+x y=x^3+a^{223} x^2+a^{57}$
- $y^2+x y=x^3+a^{38} x^2+a^{111}$
- $y^2+x y=x^3+a^{218} x^2+a^{89}$
- $y^2+x y=x^3+a^{149} x^2+a^{213}$
- $y^2+x y=x^3+a^{207} x^2+a^{39}$
- $y^2+x y=x^3+a^{15} x^2+a^{83}$
- $y^2+x y=x^3+a^{24} x^2+a^{145}$
- $y^2+x y=x^3+a^{192} x^2+a^{83}$
- $y^2+x y=x^3+a^{133} x^2+a^{99}$
- $y^2+x y=x^3+a^{51} x^2+a^{179}$
- $y^2+x y=x^3+a^{174} x^2+a^9$
- $y^2+x y=x^3+a^9 x^2+a^{205}$
- $y^2+x y=x^3+a^{174} x^2+a^{117}$
- $y^2+x y=x^3+a^{50} x^2+a^5$
- $y^2+x y=x^3+x^2+a^{13}$
- $y^2+x y=x^3+a^{238} x^2+a^{15}$
- $y^2+x y=x^3+a^{149} x^2+a^{143}$
- $y^2+x y=x^3+a^{99} x^2+a^{183}$
- $y^2+x y=x^3+a^{203} x^2+a^{181}$
- $y^2+x y=x^3+a^{23} x^2+a^{211}$
where $a$ is a root of the Conway polynomial.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3^{5}}$.
Endomorphism algebra over $\F_{3^{5}}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-194}) \). |
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 |
|---|---|---|
| 1.243.o | $2$ | (not in LMFDB) |