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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 463680r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
463680.r4 | 463680r1 | \([0, 0, 0, 1332, -32581008]\) | \(1367631/2399636575\) | \(-458577690800947200\) | \([2]\) | \(5898240\) | \(2.0678\) | \(\Gamma_0(N)\)-optimal* |
463680.r3 | 463680r2 | \([0, 0, 0, -1522188, -710852112]\) | \(2041085246738049/38897700625\) | \(7433466348994560000\) | \([2, 2]\) | \(11796480\) | \(2.4144\) | \(\Gamma_0(N)\)-optimal* |
463680.r2 | 463680r3 | \([0, 0, 0, -3178188, 1111410288]\) | \(18577831198352049/7958740140575\) | \(1520938926074540851200\) | \([2]\) | \(23592960\) | \(2.7609\) | \(\Gamma_0(N)\)-optimal* |
463680.r1 | 463680r4 | \([0, 0, 0, -24242508, -45942465168]\) | \(8244966675515989329/3081640625\) | \(588910694400000000\) | \([2]\) | \(23592960\) | \(2.7609\) |
Rank
sage: E.rank()
The elliptic curves in class 463680r have rank \(0\).
Complex multiplication
The elliptic curves in class 463680r do not have complex multiplication.Modular form 463680.2.a.r
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.