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SageMath
E = EllipticCurve("cs1")
E.isogeny_class()
Elliptic curves in class 54150.cs
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 54150.cs1 | 54150cn4 | \([1, 0, 0, -476731591313, 126694824849573117]\) | \(16300610738133468173382620881/2228489100\) | \(1638144265756204687500\) | \([2]\) | \(207360000\) | \(4.8858\) | |
| 54150.cs2 | 54150cn3 | \([1, 0, 0, -29795721813, 1979605144945617]\) | \(-3979640234041473454886161/1471455901872240\) | \(-1081655300878579378803750000\) | \([2]\) | \(103680000\) | \(4.5392\) | |
| 54150.cs3 | 54150cn2 | \([1, 0, 0, -793703813, 7415066935617]\) | \(75224183150104868881/11219310000000000\) | \(8247223799407968750000000000\) | \([2]\) | \(41472000\) | \(4.0810\) | |
| 54150.cs4 | 54150cn1 | \([1, 0, 0, 84248187, 632887735617]\) | \(89962967236397039/287450726400000\) | \(-211302697930905600000000000\) | \([2]\) | \(20736000\) | \(3.7345\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 54150.cs have rank \(0\).
Complex multiplication
The elliptic curves in class 54150.cs do not have complex multiplication.Modular form 54150.2.a.cs
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
