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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 1575.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1575.e1 | 1575j1 | \([1, -1, 1, -4055, -95178]\) | \(5177717/189\) | \(269103515625\) | \([2]\) | \(1920\) | \(0.96281\) | \(\Gamma_0(N)\)-optimal |
| 1575.e2 | 1575j2 | \([1, -1, 1, 1570, -342678]\) | \(300763/35721\) | \(-50860564453125\) | \([2]\) | \(3840\) | \(1.3094\) |
Rank
sage: E.rank()
The elliptic curves in class 1575.e have rank \(0\).
Complex multiplication
The elliptic curves in class 1575.e do not have complex multiplication.Modular form 1575.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
