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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 85800.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 85800.f1 | 85800bx4 | \([0, -1, 0, -1743808, -885748388]\) | \(36652193922790372/93308787\) | \(1492940592000000\) | \([2]\) | \(1376256\) | \(2.1492\) | |
| 85800.f2 | 85800bx2 | \([0, -1, 0, -110308, -13459388]\) | \(37109806448848/1803785841\) | \(7215143364000000\) | \([2, 2]\) | \(688128\) | \(1.8026\) | |
| 85800.f3 | 85800bx1 | \([0, -1, 0, -19183, 756112]\) | \(3122884507648/835956693\) | \(208989173250000\) | \([2]\) | \(344064\) | \(1.4561\) | \(\Gamma_0(N)\)-optimal |
| 85800.f4 | 85800bx3 | \([0, -1, 0, 65192, -52420388]\) | \(1915049403068/75239967231\) | \(-1203839475696000000\) | \([2]\) | \(1376256\) | \(2.1492\) |
Rank
sage: E.rank()
The elliptic curves in class 85800.f have rank \(1\).
Complex multiplication
The elliptic curves in class 85800.f do not have complex multiplication.Modular form 85800.2.a.f
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.
