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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 471510e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
471510.e2 | 471510e1 | \([1, -1, 0, -119430, -491724]\) | \(53540005609/30950400\) | \(108906526900454400\) | \([2]\) | \(6021120\) | \(1.9586\) | \(\Gamma_0(N)\)-optimal |
471510.e1 | 471510e2 | \([1, -1, 0, -1336230, -592586604]\) | \(74985951512809/233868960\) | \(822924943891558560\) | \([2]\) | \(12042240\) | \(2.3052\) |
Rank
sage: E.rank()
The elliptic curves in class 471510e have rank \(2\).
Complex multiplication
The elliptic curves in class 471510e do not have complex multiplication.Modular form 471510.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.