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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 97682e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 97682.h4 | 97682e1 | \([1, 1, 0, -147540, 13443344]\) | \(3048625/1088\) | \(126760089592605248\) | \([2]\) | \(1327104\) | \(1.9827\) | \(\Gamma_0(N)\)-optimal |
| 97682.h3 | 97682e2 | \([1, 1, 0, -2101180, 1171170408]\) | \(8805624625/2312\) | \(269365190384286152\) | \([2]\) | \(2654208\) | \(2.3293\) | |
| 97682.h2 | 97682e3 | \([1, 1, 0, -5031640, -4345713588]\) | \(120920208625/19652\) | \(2289604118266432292\) | \([2]\) | \(3981312\) | \(2.5320\) | |
| 97682.h1 | 97682e4 | \([1, 1, 0, -5520050, -3451825606]\) | \(159661140625/48275138\) | \(5624412516521490925298\) | \([2]\) | \(7962624\) | \(2.8786\) |
Rank
sage: E.rank()
The elliptic curves in class 97682e have rank \(1\).
Complex multiplication
The elliptic curves in class 97682e do not have complex multiplication.Modular form 97682.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
