Show commands:
SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 10710.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 10710.e1 | 10710h3 | \([1, -1, 0, -290655, -60224099]\) | \(3725035528036823281/1203203526000\) | \(877135370454000\) | \([2]\) | \(98304\) | \(1.8416\) | |
| 10710.e2 | 10710h2 | \([1, -1, 0, -20655, -662099]\) | \(1336852858103281/509796000000\) | \(371641284000000\) | \([2, 2]\) | \(49152\) | \(1.4950\) | |
| 10710.e3 | 10710h1 | \([1, -1, 0, -9135, 330925]\) | \(115650783909361/2924544000\) | \(2131992576000\) | \([2]\) | \(24576\) | \(1.1485\) | \(\Gamma_0(N)\)-optimal |
| 10710.e4 | 10710h4 | \([1, -1, 0, 65025, -4791875]\) | \(41709358422320399/37652343750000\) | \(-27448558593750000\) | \([2]\) | \(98304\) | \(1.8416\) |
Rank
sage: E.rank()
The elliptic curves in class 10710.e have rank \(1\).
Complex multiplication
The elliptic curves in class 10710.e do not have complex multiplication.Modular form 10710.2.a.e
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
