Invariants
| Base field: | $\F_{31}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 29 x^{2} + 961 x^{4}$ |
| Frobenius angles: | $\pm0.327466167327$, $\pm0.672533832673$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{33}, \sqrt{-91})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $28$ |
| Isomorphism classes: | 32 |
This isogeny class is simple but not geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
| $p$-rank: | $2$ |
| Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $991$ | $982081$ | $887444464$ | $854890707609$ | $819628324213351$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $32$ | $1020$ | $29792$ | $925684$ | $28629152$ | $887385246$ | $27512614112$ | $852892394404$ | $26439622160672$ | $819628361445900$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 28 curves (of which all are hyperelliptic):
- $y^2=7 x^6+3 x^5+7 x^4+28 x^3+29 x^2+30 x+16$
- $y^2=21 x^6+9 x^5+21 x^4+22 x^3+25 x^2+28 x+17$
- $y^2=12 x^6+7 x^5+5 x^4+2 x^3+9 x^2+19 x+23$
- $y^2=5 x^6+21 x^5+15 x^4+6 x^3+27 x^2+26 x+7$
- $y^2=16 x^6+11 x^5+12 x^4+8 x^3+14 x^2+21 x+30$
- $y^2=17 x^6+18 x^5+15 x^4+12 x^3+30 x^2+28 x+30$
- $y^2=20 x^6+23 x^5+14 x^4+5 x^3+28 x^2+22 x+28$
- $y^2=23 x^6+22 x^5+29 x^4+22 x^3+x^2+21 x+1$
- $y^2=10 x^6+7 x^5+14 x^4+7 x^3+3 x^2+28 x+13$
- $y^2=26 x^6+11 x^5+24 x^4+25 x^3+3 x^2+18 x+15$
- $y^2=16 x^6+2 x^5+10 x^4+13 x^3+9 x^2+23 x+14$
- $y^2=x^6+15 x^5+7 x^4+10 x^3+20 x^2+28 x+16$
- $y^2=3 x^6+14 x^5+21 x^4+30 x^3+29 x^2+22 x+17$
- $y^2=4 x^6+18 x^5+15 x^4+16 x^3+20 x^2+11 x+14$
- $y^2=12 x^6+23 x^5+14 x^4+17 x^3+29 x^2+2 x+11$
- $y^2=16 x^6+10 x^5+18 x^4+11 x^3+22 x^2+13 x+16$
- $y^2=17 x^6+30 x^5+23 x^4+2 x^3+4 x^2+8 x+17$
- $y^2=24 x^6+3 x^5+30 x^4+7 x^3+23 x^2+27 x+19$
- $y^2=10 x^6+9 x^5+28 x^4+21 x^3+7 x^2+19 x+26$
- $y^2=2 x^6+29 x^5+19 x^4+8 x^3+11 x^2+11 x+12$
- $y^2=6 x^6+25 x^5+26 x^4+24 x^3+2 x^2+2 x+5$
- $y^2=9 x^6+8 x^5+18 x^4+2 x^3+26 x^2+x+21$
- $y^2=9 x^6+2 x^5+18 x^4+30 x^3+20 x^2+10 x+16$
- $y^2=27 x^6+6 x^5+23 x^4+28 x^3+29 x^2+30 x+17$
- $y^2=14 x^6+15 x^5+4 x^4+18 x^3+16 x^2+21 x+18$
- $y^2=11 x^6+14 x^5+12 x^4+23 x^3+17 x^2+x+23$
- $y^2=6 x^6+7 x^5+10 x^4+4 x^3+3 x^2+15 x+27$
- $y^2=18 x^6+21 x^5+30 x^4+12 x^3+9 x^2+14 x+19$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{31^{2}}$.
Endomorphism algebra over $\F_{31}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{33}, \sqrt{-91})\). |
| The base change of $A$ to $\F_{31^{2}}$ is 1.961.bd 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3003}) \)$)$ |
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 |
|---|---|---|
| 2.31.a_abd | $4$ | (not in LMFDB) |