Invariants
Base field: | $\F_{37}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 19 x + 161 x^{2} - 703 x^{3} + 1369 x^{4}$ |
Frobenius angles: | $\pm0.120598710666$, $\pm0.281943368926$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.303693.1 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $809$ | $1822677$ | $2576338973$ | $3516922710549$ | $4809274924231424$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $19$ | $1331$ | $50863$ | $1876531$ | $69353914$ | $2565734483$ | $94931806891$ | $3512480575459$ | $129961764458959$ | $4808584600031366$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=12x^6+8x^5+30x^4+12x^3+27x^2+18x+23$
- $y^2=32x^6+3x^5+x^4+17x^3+23x^2+x+15$
- $y^2=15x^5+23x^4+21x^3+25x^2+7x+25$
- $y^2=13x^6+3x^5+x^4+4x^3+28x^2+6x+14$
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 4.0.303693.1. |
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.t_gf | $2$ | (not in LMFDB) |