Invariants
| Base field: | $\F_{31}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 14 x + 100 x^{2} - 434 x^{3} + 961 x^{4}$ |
| Frobenius angles: | $\pm0.122835614637$, $\pm0.392689253259$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.3755840.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $8$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $614$ | $927140$ | $892110686$ | $852453310160$ | $819431507511054$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $18$ | $966$ | $29946$ | $923046$ | $28622278$ | $887518038$ | $27513147918$ | $852894549246$ | $26439630781746$ | $819628274371126$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=24 x^6+x^5+5 x^4+30 x^3+23 x^2+13 x+15$
- $y^2=23 x^6+17 x^5+13 x^4+21 x^2+17 x+29$
- $y^2=27 x^6+12 x^5+25 x^4+22 x^3+29 x^2+10 x+17$
- $y^2=12 x^6+26 x^5+12 x^4+4 x^3+5 x^2+24 x+11$
- $y^2=21 x^6+17 x^5+23 x^4+30 x^3+21 x^2+3 x$
- $y^2=24 x^6+21 x^5+14 x^4+22 x^3+19 x^2+5 x+6$
- $y^2=21 x^6+9 x^5+18 x^4+21 x^3+6 x^2+27 x+24$
- $y^2=11 x^6+25 x^5+16 x^4+16 x^3+22 x^2+25 x+7$
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 4.0.3755840.1. |
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_dw | $2$ | (not in LMFDB) |