Invariants
| Base field: | $\F_{37}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 18 x + 153 x^{2} - 666 x^{3} + 1369 x^{4}$ |
| Frobenius angles: | $\pm0.172915768399$, $\pm0.285690670341$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.4672.2 |
| Galois group: | $D_{4}$ |
| Jacobians: | $7$ |
| Isomorphism classes: | 7 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $839$ | $1851673$ | $2587720988$ | $3520017411289$ | $4809893920031999$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $20$ | $1352$ | $51086$ | $1878180$ | $69362840$ | $2565763886$ | $94931747336$ | $3512478458500$ | $129961740846134$ | $4808584405886312$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 7 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=30x^6+15x^5+34x^4+13x^3+30x^2+36x+25$
- $y^2=13x^6+10x^5+7x^4+13x^3+34x^2+32x+9$
- $y^2=13x^6+35x^5+30x^4+35x^3+34x^2+23x+13$
- $y^2=8x^6+32x^5+16x^4+22x^3+13x^2+11x+22$
- $y^2=8x^6+12x^5+34x^4+3x^3+29x^2+36x+31$
- $y^2=32x^6+18x^5+14x^4+32x^2+x+34$
- $y^2=17x^6+25x^5+5x^4+22x^3+13x^2+x+4$
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 4.0.4672.2. |
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_fx | $2$ | (not in LMFDB) |