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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 5746e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 5746.b4 | 5746e1 | \([1, 0, 1, -511, 2706]\) | \(3048625/1088\) | \(5251568192\) | \([2]\) | \(4608\) | \(0.56608\) | \(\Gamma_0(N)\)-optimal |
| 5746.b3 | 5746e2 | \([1, 0, 1, -7271, 237954]\) | \(8805624625/2312\) | \(11159582408\) | \([2]\) | \(9216\) | \(0.91266\) | |
| 5746.b2 | 5746e3 | \([1, 0, 1, -17411, -885558]\) | \(120920208625/19652\) | \(94856450468\) | \([2]\) | \(13824\) | \(1.1154\) | |
| 5746.b1 | 5746e4 | \([1, 0, 1, -19101, -703714]\) | \(159661140625/48275138\) | \(233014870574642\) | \([2]\) | \(27648\) | \(1.4620\) |
Rank
sage: E.rank()
The elliptic curves in class 5746e have rank \(1\).
Complex multiplication
The elliptic curves in class 5746e do not have complex multiplication.Modular form 5746.2.a.e
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 Cremona 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 Cremona labels.
