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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 1225.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1225.e1 | 1225a3 | \([0, 1, 1, -160883, 25929019]\) | \(-250523582464/13671875\) | \(-25132537841796875\) | \([]\) | \(6912\) | \(1.9051\) | |
| 1225.e2 | 1225a1 | \([0, 1, 1, -1633, -28731]\) | \(-262144/35\) | \(-64339296875\) | \([]\) | \(768\) | \(0.80652\) | \(\Gamma_0(N)\)-optimal |
| 1225.e3 | 1225a2 | \([0, 1, 1, 10617, 75394]\) | \(71991296/42875\) | \(-78815638671875\) | \([]\) | \(2304\) | \(1.3558\) |
Rank
sage: E.rank()
The elliptic curves in class 1225.e have rank \(0\).
Complex multiplication
The elliptic curves in class 1225.e do not have complex multiplication.Modular form 1225.2.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.
\(\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
