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SageMath
E = EllipticCurve("dc1")
E.isogeny_class()
Elliptic curves in class 139230.dc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 139230.dc1 | 139230bc3 | \([1, -1, 1, -6660113, 6617282721]\) | \(44816807438220995641801/9512718589920\) | \(6934771852051680\) | \([2]\) | \(3932160\) | \(2.4251\) | |
| 139230.dc2 | 139230bc4 | \([1, -1, 1, -810833, -121049823]\) | \(80870462846141298121/38087635627860000\) | \(27765886372709940000\) | \([2]\) | \(3932160\) | \(2.4251\) | |
| 139230.dc3 | 139230bc2 | \([1, -1, 1, -417713, 102714081]\) | \(11056793118237203401/159353257190400\) | \(116168524491801600\) | \([2, 2]\) | \(1966080\) | \(2.0785\) | |
| 139230.dc4 | 139230bc1 | \([1, -1, 1, -2993, 4342497]\) | \(-4066120948681/11168482590720\) | \(-8141823808634880\) | \([2]\) | \(983040\) | \(1.7319\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 139230.dc have rank \(1\).
Complex multiplication
The elliptic curves in class 139230.dc do not have complex multiplication.Modular form 139230.2.a.dc
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
