Invariants
| Base field: | $\F_{31}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 9 x + 78 x^{2} - 279 x^{3} + 961 x^{4}$ |
| Frobenius angles: | $\pm0.299425345194$, $\pm0.429727371569$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.690421.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $26$ |
| Isomorphism classes: | 26 |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
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: | $2$ |
| Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $752$ | $998656$ | $903720512$ | $853207745536$ | $819357540167312$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $23$ | $1037$ | $30332$ | $923865$ | $28619693$ | $887470382$ | $27512628971$ | $852890799889$ | $26439618663092$ | $819628310178677$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 26 curves (of which all are hyperelliptic):
- $y^2=16 x^6+26 x^5+12 x^4+25 x^3+11 x^2+27$
- $y^2=15 x^5+21 x^4+25 x^3+12 x^2+11 x$
- $y^2=11 x^6+22 x^5+2 x^4+20 x^3+4 x^2+22 x+29$
- $y^2=22 x^6+2 x^5+23 x^4+26 x^3+5 x^2+15 x+27$
- $y^2=23 x^6+21 x^5+24 x^4+25 x^3+21 x^2+13 x+1$
- $y^2=18 x^6+10 x^5+29 x^4+12 x^3+22 x^2+14 x+22$
- $y^2=16 x^6+21 x^5+30 x^4+6 x^3+13 x^2+29 x+27$
- $y^2=27 x^6+15 x^5+15 x^4+6 x^3+23 x^2+9 x+20$
- $y^2=12 x^6+13 x^5+13 x^4+29 x^3+15 x^2+25 x+18$
- $y^2=6 x^6+x^5+16 x^4+3 x^3+26 x^2+14 x+8$
- $y^2=23 x^6+30 x^5+12 x^4+14 x^3+20 x^2+17 x+11$
- $y^2=12 x^6+20 x^5+2 x^4+30 x^3+x^2+26 x+21$
- $y^2=16 x^6+28 x^5+30 x^4+16 x^3+20 x^2+13 x+30$
- $y^2=13 x^6+9 x^5+23 x^4+24 x^3+5 x^2+27 x+19$
- $y^2=22 x^6+23 x^4+10 x^3+2 x^2+23$
- $y^2=10 x^6+18 x^5+x^4+28 x^3+5 x^2+10 x+21$
- $y^2=14 x^6+6 x^5+22 x^4+19 x^3+10 x^2+2 x+12$
- $y^2=30 x^5+14 x^4+7 x^3+5 x^2+3 x+3$
- $y^2=30 x^6+14 x^5+20 x^4+24 x^3+9 x^2+26 x+7$
- $y^2=x^6+15 x^5+8 x^4+6 x^3+2 x^2+30 x+12$
- $y^2=6 x^6+27 x^5+27 x^4+11 x^3+21 x^2+2 x+2$
- $y^2=26 x^6+14 x^5+18 x^4+9 x^3+11 x^2+x$
- $y^2=29 x^6+19 x^5+24 x^4+20 x^3+7 x^2+7 x+26$
- $y^2=17 x^6+29 x^5+15 x^4+16 x^3+26 x^2+6 x+6$
- $y^2=3 x^6+27 x^5+17 x^4+20 x^3+3 x^2+22 x+28$
- $y^2=23 x^6+11 x^5+x^4+11 x^3+x^2+12 x+26$
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 4.0.690421.1. |
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.j_da | $2$ | (not in LMFDB) |