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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 60333.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 60333.k1 | 60333e4 | \([1, 1, 0, -4183091, -3294761400]\) | \(1677087406638588673/4641\) | \(22401220569\) | \([2]\) | \(860160\) | \(2.1044\) | |
| 60333.k2 | 60333e2 | \([1, 1, 0, -261446, -51560985]\) | \(409460675852593/21538881\) | \(103964064660729\) | \([2, 2]\) | \(430080\) | \(1.7578\) | |
| 60333.k3 | 60333e3 | \([1, 1, 0, -247081, -57459254]\) | \(-345608484635233/94427721297\) | \(-455784575005851273\) | \([2]\) | \(860160\) | \(2.1044\) | |
| 60333.k4 | 60333e1 | \([1, 1, 0, -17241, -717504]\) | \(117433042273/22801233\) | \(110057196655497\) | \([2]\) | \(215040\) | \(1.4113\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 60333.k have rank \(1\).
Complex multiplication
The elliptic curves in class 60333.k do not have complex multiplication.Modular form 60333.2.a.k
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.
