Invariants
| Base field: | $\F_{37}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 17 x + 143 x^{2} - 629 x^{3} + 1369 x^{4}$ |
| Frobenius angles: | $\pm0.178476432488$, $\pm0.314433878567$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.729573.2 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $867$ | $1871853$ | $2590822287$ | $3519220285269$ | $4809414360773712$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $21$ | $1367$ | $51147$ | $1877755$ | $69355926$ | $2565730271$ | $94931816427$ | $3512480478835$ | $129961751944077$ | $4808584388917862$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=13x^6+32x^5+35x^4+10x^3+10x^2+21x+18$
- $y^2=31x^6+36x^5+10x^4+28x^3+19x^2+22x+21$
- $y^2=19x^6+x^5+6x^4+27x^3+30x^2+21x+15$
- $y^2=35x^6+7x^5+22x^4+29x^3+32x^2+24x+14$
- $y^2=10x^6+23x^5+27x^4+21x^3+6x^2+32x+10$
- $y^2=4x^6+31x^5+33x^4+12x^3+28x^2+24x+31$
- $y^2=24x^6+30x^5+32x^4+3x^3+20x^2+3x+20$
- $y^2=2x^6+x^5+35x^4+7x^3+21x^2+20x+30$
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 4.0.729573.2. |
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.r_fn | $2$ | (not in LMFDB) |