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SageMath
E = EllipticCurve("df1")
E.isogeny_class()
Elliptic curves in class 463680df
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
463680.df4 | 463680df1 | \([0, 0, 0, 16692, -1920112]\) | \(2691419471/9891840\) | \(-1890360062115840\) | \([2]\) | \(2359296\) | \(1.6142\) | \(\Gamma_0(N)\)-optimal* |
463680.df3 | 463680df2 | \([0, 0, 0, -167628, -23080048]\) | \(2725812332209/373262400\) | \(71331555468902400\) | \([2, 2]\) | \(4718592\) | \(1.9607\) | \(\Gamma_0(N)\)-optimal* |
463680.df2 | 463680df3 | \([0, 0, 0, -697548, 200970128]\) | \(196416765680689/22365315000\) | \(4274078255677440000\) | \([2]\) | \(9437184\) | \(2.3073\) | \(\Gamma_0(N)\)-optimal* |
463680.df1 | 463680df4 | \([0, 0, 0, -2586828, -1601366128]\) | \(10017490085065009/235066440\) | \(44921896241725440\) | \([2]\) | \(9437184\) | \(2.3073\) |
Rank
sage: E.rank()
The elliptic curves in class 463680df have rank \(1\).
Complex multiplication
The elliptic curves in class 463680df do not have complex multiplication.Modular form 463680.2.a.df
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.