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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 64974r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 64974.x4 | 64974r1 | \([1, 0, 1, 415, -5932]\) | \(67419143/169728\) | \(-19968329472\) | \([2]\) | \(49152\) | \(0.66026\) | \(\Gamma_0(N)\)-optimal |
| 64974.x3 | 64974r2 | \([1, 0, 1, -3505, -67084]\) | \(40459583737/7033104\) | \(827437652496\) | \([2, 2]\) | \(98304\) | \(1.0068\) | |
| 64974.x2 | 64974r3 | \([1, 0, 1, -16245, 732988]\) | \(4029546653497/351790452\) | \(41387794887348\) | \([2]\) | \(196608\) | \(1.3534\) | |
| 64974.x1 | 64974r4 | \([1, 0, 1, -53485, -4765204]\) | \(143820170742457/5826444\) | \(685475310156\) | \([2]\) | \(196608\) | \(1.3534\) |
Rank
sage: E.rank()
The elliptic curves in class 64974r have rank \(1\).
Complex multiplication
The elliptic curves in class 64974r do not have complex multiplication.Modular form 64974.2.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.
