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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 6045.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6045.f1 | 6045h1 | \([0, 1, 1, -3773731, 2820401431]\) | \(-5943423068131740751396864/30737464875\) | \(-30737464875\) | \([3]\) | \(58320\) | \(2.0881\) | \(\Gamma_0(N)\)-optimal |
| 6045.f2 | 6045h2 | \([0, 1, 1, -3758071, 2844985660]\) | \(-5869738723523437004161024/102823888385232421875\) | \(-102823888385232421875\) | \([3]\) | \(174960\) | \(2.6374\) | |
| 6045.f3 | 6045h3 | \([0, 1, 1, 14606639, 13595424541]\) | \(344647053641493631661244416/279240310192108154296875\) | \(-279240310192108154296875\) | \([]\) | \(524880\) | \(3.1867\) |
Rank
sage: E.rank()
The elliptic curves in class 6045.f have rank \(0\).
Complex multiplication
The elliptic curves in class 6045.f do not have complex multiplication.Modular form 6045.2.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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
