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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 36225.z
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 36225.z1 | 36225k1 | \([1, -1, 1, -103730, -12507728]\) | \(292583028222603/8456021875\) | \(3567384228515625\) | \([2]\) | \(207360\) | \(1.7622\) | \(\Gamma_0(N)\)-optimal |
| 36225.z2 | 36225k2 | \([1, -1, 1, 24895, -41576978]\) | \(4044759171237/1771943359375\) | \(-747538604736328125\) | \([2]\) | \(414720\) | \(2.1087\) |
Rank
sage: E.rank()
The elliptic curves in class 36225.z have rank \(1\).
Complex multiplication
The elliptic curves in class 36225.z do not have complex multiplication.Modular form 36225.2.a.z
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
