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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 158400.bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 158400.bc1 | 158400cc4 | \([0, 0, 0, -6390300, -6217702000]\) | \(154639330142416/33275\) | \(6209913600000000\) | \([2]\) | \(3981312\) | \(2.4151\) | |
| 158400.bc2 | 158400cc3 | \([0, 0, 0, -400800, -96433000]\) | \(610462990336/8857805\) | \(103317437520000000\) | \([2]\) | \(1990656\) | \(2.0685\) | |
| 158400.bc3 | 158400cc2 | \([0, 0, 0, -90300, -5902000]\) | \(436334416/171875\) | \(32076000000000000\) | \([2]\) | \(1327104\) | \(1.8657\) | |
| 158400.bc4 | 158400cc1 | \([0, 0, 0, -40800, 3107000]\) | \(643956736/15125\) | \(176418000000000\) | \([2]\) | \(663552\) | \(1.5192\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 158400.bc have rank \(2\).
Complex multiplication
The elliptic curves in class 158400.bc do not have complex multiplication.Modular form 158400.2.a.bc
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
