Invariants
| Base field: | $\F_{31}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 6 x + 53 x^{2} - 186 x^{3} + 961 x^{4}$ |
| Frobenius angles: | $\pm0.274597802288$, $\pm0.535595112270$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.560704.4 |
| Galois group: | $D_{4}$ |
| Jacobians: | $27$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $823$ | $993361$ | $892892452$ | $852980216841$ | $819909906920023$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $26$ | $1032$ | $29972$ | $923620$ | $28638986$ | $887523702$ | $27512058278$ | $852888376324$ | $26439629926412$ | $819628351990152$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 27 curves (of which all are hyperelliptic):
- $y^2=29 x^6+24 x^5+3 x^4+7 x^3+6 x^2+20$
- $y^2=8 x^6+28 x^5+6 x^4+29 x^3+x^2+17 x+19$
- $y^2=12 x^6+29 x^5+11 x^4+30 x^3+2 x^2+24 x+3$
- $y^2=22 x^6+9 x^5+11 x^4+30 x^3+16 x^2+10 x+22$
- $y^2=11 x^6+23 x^5+13 x^3+x^2+3 x+30$
- $y^2=12 x^6+21 x^5+20 x^4+22 x^3+7 x^2+23 x+29$
- $y^2=14 x^6+16 x^4+28 x^3+10 x^2+18 x+25$
- $y^2=13 x^6+14 x^5+2 x^4+30 x^3+28 x^2+2 x+24$
- $y^2=21 x^6+18 x^5+13 x^4+9 x^3+18 x^2+25 x+24$
- $y^2=13 x^6+28 x^5+23 x^4+26 x^3+3 x^2+12 x+11$
- $y^2=19 x^6+12 x^5+3 x^4+x^3+4 x^2+15 x+23$
- $y^2=18 x^6+10 x^5+21 x^4+11 x^3+26 x^2+6 x+18$
- $y^2=11 x^6+29 x^5+18 x^4+4 x^3+7 x^2+19 x+12$
- $y^2=14 x^6+12 x^5+3 x^4+4 x^3+22 x^2+7 x+22$
- $y^2=26 x^6+27 x^5+9 x^3+3 x^2+7 x+20$
- $y^2=8 x^6+7 x^5+15 x^4+13 x^3+5 x^2+16 x+4$
- $y^2=27 x^6+22 x^5+19 x^4+x^3+30 x^2+11 x+24$
- $y^2=9 x^6+5 x^5+3 x^4+11 x^3+5 x^2+28 x+24$
- $y^2=17 x^6+4 x^5+19 x^4+18 x^3+27 x^2+11 x+19$
- $y^2=19 x^6+7 x^5+19 x^4+15 x^3+3 x^2+19 x+27$
- $y^2=18 x^6+21 x^5+8 x^4+15 x^3+6 x^2+20 x+15$
- $y^2=9 x^6+15 x^5+21 x^4+25 x^3+17 x^2+8 x+19$
- $y^2=9 x^6+9 x^5+11 x^4+6 x^3+22 x^2+28 x+13$
- $y^2=27 x^6+2 x^5+29 x^4+6 x^3+26 x^2+19 x+1$
- $y^2=26 x^6+5 x^5+16 x^4+11 x^3+25 x^2+11 x+20$
- $y^2=23 x^6+20 x^5+23 x^4+10 x^3+19 x^2+5 x+18$
- $y^2=11 x^6+x^5+15 x^4+19 x^3+27 x^2+25 x+17$
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 4.0.560704.4. |
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.g_cb | $2$ | (not in LMFDB) |