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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 2178.a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 2178.a1 | 2178a4 | \([1, -1, 0, -160287, -6133231]\) | \(13060888875/7086244\) | \(247094742955997772\) | \([2]\) | \(23040\) | \(2.0283\) | |
| 2178.a2 | 2178a2 | \([1, -1, 0, -123987, -16773003]\) | \(4406910829875/7744\) | \(370412146368\) | \([2]\) | \(7680\) | \(1.4790\) | |
| 2178.a3 | 2178a3 | \([1, -1, 0, -94947, 11208005]\) | \(2714704875/21296\) | \(742583750431248\) | \([2]\) | \(11520\) | \(1.6817\) | |
| 2178.a4 | 2178a1 | \([1, -1, 0, -7827, -255051]\) | \(1108717875/45056\) | \(2155125215232\) | \([2]\) | \(3840\) | \(1.1324\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 2178.a have rank \(0\).
Complex multiplication
The elliptic curves in class 2178.a do not have complex multiplication.Modular form 2178.2.a.a
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 2 & 6 \\ 3 & 1 & 6 & 2 \\ 2 & 6 & 1 & 3 \\ 6 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
