Invariants
| Base field: | $\F_{31}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 7 x + 63 x^{2} - 217 x^{3} + 961 x^{4}$ |
| Frobenius angles: | $\pm0.288947435307$, $\pm0.495829374926$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.238725.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $24$ |
| Isomorphism classes: | 30 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $801$ | $1000449$ | $897369111$ | $852683683149$ | $819661209210576$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $25$ | $1039$ | $30121$ | $923299$ | $28630300$ | $887523163$ | $27512313595$ | $852888169939$ | $26439624144871$ | $819628397589454$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 24 curves (of which all are hyperelliptic):
- $y^2=21 x^6+19 x^5+15 x^4+10 x^3+5 x^2+17 x+30$
- $y^2=8 x^6+17 x^5+4 x^4+13 x^3+23 x$
- $y^2=23 x^6+21 x^5+12 x^4+5 x^3+10 x^2+16 x+21$
- $y^2=3 x^6+9 x^5+8 x^4+29 x^3+12 x^2+9 x+15$
- $y^2=19 x^6+27 x^5+12 x^4+23 x^3+23 x^2+22 x+21$
- $y^2=16 x^6+14 x^5+8 x^4+11 x^3+10 x^2+20 x+27$
- $y^2=2 x^6+29 x^5+18 x^4+5 x^3+25 x^2+12 x+7$
- $y^2=23 x^6+24 x^5+21 x^4+16 x^3+27 x^2+21 x+11$
- $y^2=23 x^6+8 x^5+30 x^4+16 x^3+19 x^2+13 x+16$
- $y^2=11 x^6+10 x^5+16 x^4+25 x^3+5 x^2+19 x+10$
- $y^2=5 x^6+18 x^5+19 x^4+25 x^3+5 x^2+9 x+29$
- $y^2=15 x^6+11 x^5+2 x^4+20 x^3+8 x^2+5 x+25$
- $y^2=20 x^6+25 x^5+17 x^4+30 x^3+11 x^2+6 x+12$
- $y^2=12 x^6+30 x^5+15 x^4+26 x^3+5 x^2+23 x+11$
- $y^2=27 x^6+10 x^5+28 x^4+18 x^3+28 x^2+27 x+6$
- $y^2=28 x^6+9 x^5+10 x^3+23 x^2+29 x+29$
- $y^2=4 x^6+4 x^5+17 x^4+8 x^3+13 x^2+1$
- $y^2=24 x^6+5 x^5+29 x^4+12 x^3+4 x+22$
- $y^2=24 x^6+19 x^5+14 x^4+19 x^3+22 x^2+14 x+1$
- $y^2=10 x^6+13 x^5+19 x^4+3 x^3+28 x^2+12 x+8$
- $y^2=7 x^6+24 x^5+11 x^4+23 x^3+4 x^2+13$
- $y^2=3 x^6+22 x^5+18 x^3+26 x^2+26 x+28$
- $y^2=29 x^6+6 x^5+25 x^4+14 x^3+14 x^2+2 x+3$
- $y^2=26 x^6+x^5+19 x^4+19 x^3+10 x^2+14 x+23$
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 4.0.238725.1. |
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.h_cl | $2$ | (not in LMFDB) |