Invariants
Base field: | $\F_{37}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 18 x + 152 x^{2} - 666 x^{3} + 1369 x^{4}$ |
Frobenius angles: | $\pm0.156082538435$, $\pm0.296191855911$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.449856.2 |
Galois group: | $D_{4}$ |
Jacobians: | $8$ |
Isomorphism classes: | 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $838$ | $1848628$ | $2584955974$ | $3518730296784$ | $4809544382962918$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $20$ | $1350$ | $51032$ | $1877494$ | $69357800$ | $2565748230$ | $94931867540$ | $3512480621854$ | $129961757858564$ | $4808584483773750$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=9x^6+6x^5+36x^4+32x^2+3x+8$
- $y^2=4x^6+4x^5+20x^4+20x^3+4x^2+21x+32$
- $y^2=11x^6+11x^5+30x^4+17x^3+9x^2+8x+17$
- $y^2=24x^6+14x^5+36x^4+5x^3+28x^2+25x+11$
- $y^2=27x^6+27x^5+10x^4+17x^3+18x^2+2x+15$
- $y^2=24x^6+12x^5+32x^4+14x^3+24x^2+20x+17$
- $y^2=2x^6+7x^5+3x^4+2x^3+6x^2+7x+35$
- $y^2=x^6+33x^5+10x^4+9x^3+34x^2+21x+16$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{37}$.
Endomorphism algebra over $\F_{37}$The endomorphism algebra of this simple isogeny class is 4.0.449856.2. |
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 |
---|---|---|
2.37.s_fw | $2$ | (not in LMFDB) |