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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 6336.x
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6336.x1 | 6336bd3 | \([0, 0, 0, -84396, -9427664]\) | \(347873904937/395307\) | \(75544344133632\) | \([2]\) | \(24576\) | \(1.5764\) | |
| 6336.x2 | 6336bd2 | \([0, 0, 0, -6636, -65360]\) | \(169112377/88209\) | \(16857002409984\) | \([2, 2]\) | \(12288\) | \(1.2298\) | |
| 6336.x3 | 6336bd1 | \([0, 0, 0, -3756, 87856]\) | \(30664297/297\) | \(56757583872\) | \([2]\) | \(6144\) | \(0.88325\) | \(\Gamma_0(N)\)-optimal |
| 6336.x4 | 6336bd4 | \([0, 0, 0, 25044, -508880]\) | \(9090072503/5845851\) | \(-1117159523352576\) | \([2]\) | \(24576\) | \(1.5764\) |
Rank
sage: E.rank()
The elliptic curves in class 6336.x have rank \(1\).
Complex multiplication
The elliptic curves in class 6336.x do not have complex multiplication.Modular form 6336.2.a.x
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.
