Is this an elliptic curve?

Question

I am trying to learn what elliptic curves are. Sofar I have not had any luck understanding when a curve is elliptic and when it is not.

$$y^2 = \frac{x}{4}-\frac{17 x^2}{40}+\frac{23 x^3}{90}-\frac{23 x^4}{360}+\frac{x^5}{180}$$

elliptic curve or not

The complete program in Mathematica:

Clear[nn, t, n, k, M, x];
nn = 6;(*size of matrix*)
Table[
 Clear[t, n, k, M, x];
 t[n_, 1] = 1;
 t[1, k_] = 1;
 t[n_, k_] := 
  t[n, k] = 
   If[n < k, 
    If[And[n > 1, k > 1], x - Sum[t[k - i, n], {i, 1, n - 1}], 0], 
    If[And[n > 1, k > 1], x - Sum[t[n - i, k], {i, 1, k - 1}], 0]];
 M = Table[Table[t[n, k]/n, {k, 1, nn}], {n, 1, nn}];
 (*MatrixForm[M]*)
 a = Det[M]; (*polynomial*)
 xy = 5;
 xmin = -5;
 xmax = 10;
 ymin = -1;
 ymax = 1;
 {nn, ContourPlot[y^2 == a, {x, xmin, xmax}, {y, ymin, ymax}]}, {nn, 
  1, 24}]
(*Solve[Det[M]==0,x] (*solutions to the polynomial*)*)

I don't think it is. Loosely based on the definition of elliptic curves, highest power of the $x$ should be 3. — Kaster, Jul 20 '15 at 20:09
This has genus $2$, according to Maple, so it is not elliptic. — Robert Israel, Jul 20 '15 at 20:18
I don't understand what the connection is between the code and the question. — Mariano Suárez-Álvarez, Jul 20 '15 at 20:24
The polynomial is the 6-th polynomial in the code. The rest of the code is only there for my own memory. — Mats Granvik, Jul 20 '15 at 20:26
By Harnack's curve theorem, the genus of the curve should be at least 2 — hence it is not an elliptic curve. — Watson, Apr 06 '18 at 21:21

score 3 · Accepted Answer · answered Jul 20 '15 at 20:18

It is definitely not an elliptic curve: an equation $y^2=f(x)$ with $f$ having no repeated factors is "hyper-elliptic", giving an elliptic curve exactly when the degree of $f$ is $3$ or $4$. It is a curve of genus $g$ for $f$ of degree $2g+1$ and $2g+2$, generally, as one can see from the Riemann-Hurwitz formula, taking into account the points lying over $\infty$ on the Riemann sphere $\mathbb P^1\supset \mathbb C^1$.

Is this an elliptic curve?

1 Answers1