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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 2178.h
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 2178.h1 | 2178g4 | \([1, -1, 1, -1115885, 453986965]\) | \(4406910829875/7744\) | \(270030454702272\) | \([2]\) | \(23040\) | \(2.0283\) | |
| 2178.h2 | 2178g3 | \([1, -1, 1, -70445, 6956821]\) | \(1108717875/45056\) | \(1571086281904128\) | \([2]\) | \(11520\) | \(1.6817\) | |
| 2178.h3 | 2178g2 | \([1, -1, 1, -17810, 233093]\) | \(13060888875/7086244\) | \(338950264685868\) | \([2]\) | \(7680\) | \(1.4790\) | |
| 2178.h4 | 2178g1 | \([1, -1, 1, -10550, -411595]\) | \(2714704875/21296\) | \(1018633402512\) | \([2]\) | \(3840\) | \(1.1324\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 2178.h have rank \(1\).
Complex multiplication
The elliptic curves in class 2178.h do not have complex multiplication.Modular form 2178.2.a.h
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.
