Invariants
Base field: | $\F_{37}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 20 x + 172 x^{2} - 740 x^{3} + 1369 x^{4}$ |
Frobenius angles: | $\pm0.112452867781$, $\pm0.250611395750$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.84224.1 |
Galois group: | $D_{4}$ |
Jacobians: | $4$ |
Isomorphism classes: | 4 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $782$ | $1800164$ | $2570822654$ | $3516807591056$ | $4809617773833582$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $18$ | $1314$ | $50754$ | $1876470$ | $69358858$ | $2565778098$ | $94931919114$ | $3512479269534$ | $129961746737778$ | $4808584503290914$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=15x^6+34x^5+2x^4+33x^3+19x^2+29x+22$
- $y^2=6x^6+14x^5+16x^4+8x^3+21x^2+34x+36$
- $y^2=33x^6+31x^5+24x^4+10x^3+31x^2+29x+5$
- $y^2=29x^6+34x^5+28x^4+10x^3+6x^2+16x+6$
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 4.0.84224.1. |
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 |
---|---|---|
2.37.u_gq | $2$ | (not in LMFDB) |