Invariants
| Base field: | $\F_{31}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 2 x - 7 x^{2} - 62 x^{3} + 961 x^{4}$ |
| Frobenius angles: | $\pm0.181881757293$, $\pm0.730096920819$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.22030400.3 |
| Galois group: | $D_{4}$ |
| Jacobians: | $24$ |
| Isomorphism classes: | 48 |
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: | $2$ |
| Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $891$ | $907929$ | $880518276$ | $855778466169$ | $819782834354451$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $30$ | $944$ | $29556$ | $926644$ | $28634550$ | $887538998$ | $27513142890$ | $852889676644$ | $26439620142636$ | $819628272264704$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 24 curves (of which all are hyperelliptic):
- $y^2=12 x^6+2 x^4+6 x^3+29 x^2+12 x+27$
- $y^2=12 x^6+7 x^5+22 x^4+7 x^3+13 x^2+5 x+25$
- $y^2=10 x^6+5 x^5+29 x^4+10 x^3+27 x^2+27 x+5$
- $y^2=26 x^6+13 x^5+10 x^4+5 x^3+30 x^2+5 x+16$
- $y^2=9 x^6+11 x^5+5 x^4+9 x^3+13 x^2+8 x+9$
- $y^2=24 x^6+16 x^5+2 x^4+22 x^3+17 x^2+9 x+9$
- $y^2=23 x^6+18 x^5+16 x^4+27 x^3+20 x^2+28 x+18$
- $y^2=28 x^6+12 x^5+29 x^4+14 x^3+2 x^2+6 x+9$
- $y^2=28 x^6+30 x^5+x^4+6 x^3+14 x^2+13$
- $y^2=22 x^6+14 x^5+x^4+3 x^3+24 x^2+23 x+7$
- $y^2=17 x^6+24 x^5+27 x^4+7 x^3+17 x^2+29$
- $y^2=11 x^6+16 x^5+4 x^4+11 x^3+22 x^2+30 x+14$
- $y^2=25 x^6+4 x^5+27 x^4+21 x^3+8 x^2+25 x+2$
- $y^2=13 x^6+26 x^5+22 x^4+11 x^3+5 x^2+3 x+18$
- $y^2=23 x^6+24 x^5+x^4+19 x^3+28 x^2+19 x+11$
- $y^2=17 x^6+23 x^5+22 x^4+16 x^3+17 x^2+19 x+6$
- $y^2=25 x^6+29 x^5+7 x^3+11 x^2+7 x+28$
- $y^2=11 x^6+3 x^5+19 x^4+2 x^3+4 x^2+25 x+8$
- $y^2=13 x^6+25 x^5+3 x^4+8 x^3+13 x+27$
- $y^2=26 x^6+18 x^5+4 x^4+4 x^3+9 x^2+23 x+20$
- $y^2=25 x^6+14 x^5+16 x^4+16 x^3+21 x^2+x+18$
- $y^2=29 x^6+14 x^5+20 x^4+19 x^3+11 x^2+27 x+24$
- $y^2=13 x^6+2 x^5+16 x^4+16 x^3+16 x^2+2$
- $y^2=30 x^6+25 x^5+x^4+5 x^3+13 x^2+20 x+1$
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 4.0.22030400.3. |
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.c_ah | $2$ | (not in LMFDB) |