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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 36225.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 36225.k1 | 36225bf4 | \([1, -1, 1, -27830, -1779578]\) | \(209267191953/55223\) | \(629024484375\) | \([2]\) | \(81920\) | \(1.2484\) | |
| 36225.k2 | 36225bf2 | \([1, -1, 1, -1955, -20078]\) | \(72511713/25921\) | \(295256390625\) | \([2, 2]\) | \(40960\) | \(0.90182\) | |
| 36225.k3 | 36225bf1 | \([1, -1, 1, -830, 9172]\) | \(5545233/161\) | \(1833890625\) | \([2]\) | \(20480\) | \(0.55524\) | \(\Gamma_0(N)\)-optimal |
| 36225.k4 | 36225bf3 | \([1, -1, 1, 5920, -146078]\) | \(2014698447/1958887\) | \(-22312947234375\) | \([2]\) | \(81920\) | \(1.2484\) |
Rank
sage: E.rank()
The elliptic curves in class 36225.k have rank \(2\).
Complex multiplication
The elliptic curves in class 36225.k do not have complex multiplication.Modular form 36225.2.a.k
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.
