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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 340605e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
340605.e2 | 340605e1 | \([1, -1, 0, -1419, -637570]\) | \(-9/5\) | \(-175618611408645\) | \([]\) | \(906192\) | \(1.4124\) | \(\Gamma_0(N)\)-optimal |
340605.e1 | 340605e2 | \([1, -1, 0, -1704444, 860616233]\) | \(-15590912409/78125\) | \(-2744040803260078125\) | \([]\) | \(6343344\) | \(2.3853\) |
Rank
sage: E.rank()
The elliptic curves in class 340605e have rank \(1\).
Complex multiplication
The elliptic curves in class 340605e do not have complex multiplication.Modular form 340605.2.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.
\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.