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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 32760.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 32760.d1 | 32760i4 | \([0, 0, 0, -524163, 146065502]\) | \(10667565439614722/1365\) | \(2037934080\) | \([2]\) | \(163840\) | \(1.6459\) | |
| 32760.d2 | 32760i2 | \([0, 0, 0, -32763, 2281862]\) | \(5210113839844/1863225\) | \(1390890009600\) | \([2, 2]\) | \(81920\) | \(1.2993\) | |
| 32760.d3 | 32760i3 | \([0, 0, 0, -28083, 2956718]\) | \(-1640577425762/1580158125\) | \(-2359163439360000\) | \([2]\) | \(163840\) | \(1.6459\) | |
| 32760.d4 | 32760i1 | \([0, 0, 0, -2343, 24698]\) | \(7622072656/2998905\) | \(559667646720\) | \([2]\) | \(40960\) | \(0.95275\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 32760.d have rank \(1\).
Complex multiplication
The elliptic curves in class 32760.d do not have complex multiplication.Modular form 32760.2.a.d
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
