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SageMath
E = EllipticCurve("cd1")
E.isogeny_class()
Elliptic curves in class 122304cd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 122304.x4 | 122304cd1 | \([0, -1, 0, -4524, -15210]\) | \(1360251712/771147\) | \(5806379097792\) | \([2]\) | \(294912\) | \(1.1391\) | \(\Gamma_0(N)\)-optimal |
| 122304.x2 | 122304cd2 | \([0, -1, 0, -45929, 3785769]\) | \(22235451328/123201\) | \(59369367343104\) | \([2, 2]\) | \(589824\) | \(1.4857\) | |
| 122304.x3 | 122304cd3 | \([0, -1, 0, -20449, 7939009]\) | \(-245314376/6908733\) | \(-26634011564998656\) | \([2]\) | \(1179648\) | \(1.8322\) | |
| 122304.x1 | 122304cd4 | \([0, -1, 0, -733889, 242232705]\) | \(11339065490696/351\) | \(1353147973632\) | \([2]\) | \(1179648\) | \(1.8322\) |
Rank
sage: E.rank()
The elliptic curves in class 122304cd have rank \(1\).
Complex multiplication
The elliptic curves in class 122304cd do not have complex multiplication.Modular form 122304.2.a.cd
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
