Invariants
| Base field: | $\F_{31}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 15 x + 114 x^{2} - 465 x^{3} + 961 x^{4}$ |
| Frobenius angles: | $\pm0.171301927021$, $\pm0.337585460333$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-222 +2 \sqrt{17}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $12$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $596$ | $927376$ | $898293584$ | $854673431104$ | $819733498991276$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $17$ | $965$ | $30152$ | $925449$ | $28632827$ | $887503646$ | $27512744717$ | $852892857649$ | $26439631026392$ | $819628273228925$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):
- $y^2=6 x^6+26 x^5+24 x^4+7 x^3+15 x^2+10 x+8$
- $y^2=8 x^6+22 x^5+29 x^4+27 x^3+14 x^2+5 x+17$
- $y^2=29 x^6+14 x^5+24 x^4+7 x^3+23 x^2+21 x+9$
- $y^2=27 x^6+20 x^5+21 x^4+18 x^3+6 x^2+15 x+23$
- $y^2=29 x^6+2 x^5+8 x^4+4 x^2+10 x+8$
- $y^2=29 x^6+5 x^5+23 x^3+28 x+30$
- $y^2=6 x^6+20 x^5+25 x^4+27 x^3+5 x+24$
- $y^2=21 x^6+20 x^5+20 x^4+9 x^3+9 x^2+x+30$
- $y^2=11 x^6+21 x^5+13 x^4+22 x^3+7 x^2+13 x+26$
- $y^2=3 x^6+3 x^5+18 x^4+x^3+27 x+25$
- $y^2=18 x^6+24 x^5+17 x^4+16 x^3+3 x^2+19 x+27$
- $y^2=13 x^6+14 x^5+5 x^4+26 x^3+12 x^2+20 x+29$
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 \(\Q(\sqrt{-222 +2 \sqrt{17}})\). |
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.p_ek | $2$ | (not in LMFDB) |