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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 64757.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 64757.e1 | 64757i1 | \([0, -1, 1, -75129, 7951242]\) | \(-78843215872/539\) | \(-320609770019\) | \([]\) | \(161280\) | \(1.3886\) | \(\Gamma_0(N)\)-optimal |
| 64757.e2 | 64757i2 | \([0, -1, 1, -41489, 15053487]\) | \(-13278380032/156590819\) | \(-93143870995689899\) | \([]\) | \(483840\) | \(1.9380\) | |
| 64757.e3 | 64757i3 | \([0, -1, 1, 370601, -389412848]\) | \(9463555063808/115539436859\) | \(-68725551538940188739\) | \([]\) | \(1451520\) | \(2.4873\) |
Rank
sage: E.rank()
The elliptic curves in class 64757.e have rank \(1\).
Complex multiplication
The elliptic curves in class 64757.e do not have complex multiplication.Modular form 64757.2.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 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.
