Invariants
| Base field: | $\F_{31}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 14 x + 101 x^{2} - 434 x^{3} + 961 x^{4}$ |
| Frobenius angles: | $\pm0.134071731867$, $\pm0.388001791450$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.3624000.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $8$ |
| Isomorphism classes: | 16 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $615$ | $929265$ | $893371140$ | $852806005065$ | $819483595134375$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $18$ | $968$ | $29988$ | $923428$ | $28624098$ | $887521718$ | $27513147918$ | $852894585988$ | $26439630610428$ | $819628261213448$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=24 x^6+25 x^5+28 x^4+21 x^3+14 x^2+14 x+27$
- $y^2=6 x^6+19 x^5+25 x^4+18 x^3+x+18$
- $y^2=11 x^6+29 x^5+2 x^4+30 x^3+9 x^2+26 x+12$
- $y^2=6 x^6+23 x^5+7 x^4+24 x^3+5 x^2+2 x+4$
- $y^2=20 x^6+20 x^5+16 x^4+10 x^3+2 x^2+24 x+6$
- $y^2=27 x^6+28 x^5+2 x^4+14 x^3+22 x^2+27 x+3$
- $y^2=23 x^6+27 x^5+23 x^4+30 x^3+30 x^2+13 x+26$
- $y^2=13 x^6+28 x^5+4 x^4+3 x^3+8 x^2+6 x+24$
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 4.0.3624000.1. |
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.o_dx | $2$ | (not in LMFDB) |