Invariants
| Base field: | $\F_{37}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 18 x + 148 x^{2} - 666 x^{3} + 1369 x^{4}$ |
| Frobenius angles: | $\pm0.0933824788798$, $\pm0.325068141826$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.1044288.5 |
| Galois group: | $D_{4}$ |
| Jacobians: | $8$ |
| Isomorphism classes: | 8 |
This isogeny class is simple and geometrically simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
| $p$-rank: | $1$ |
| Slopes: | $[0, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $834$ | $1836468$ | $2573904978$ | $3513508539984$ | $4808021587897074$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $20$ | $1342$ | $50816$ | $1874710$ | $69335840$ | $2565645406$ | $94931763716$ | $3512483244190$ | $129961781970644$ | $4808584605976462$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=15x^6+7x^5+33x^4+28x^3+33x^2+17x+13$
- $y^2=18x^6+9x^5+13x^4+17x^3+10x^2+13$
- $y^2=16x^6+22x^5+25x^4+13x^3+27x^2+32x+19$
- $y^2=2x^6+14x^5+11x^4+x^3+25x^2+27x+17$
- $y^2=31x^6+36x^5+3x^4+35x^3+11x^2+2x+23$
- $y^2=35x^6+29x^4+30x^3+16x^2+20x+31$
- $y^2=25x^6+17x^5+11x^4+13x^3+30x^2+25x+2$
- $y^2=12x^6+25x^5+9x^4+23x^3+8x^2+25x+3$
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 4.0.1044288.5. |
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_fs | $2$ | (not in LMFDB) |