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SageMath
E = EllipticCurve("dy1")
E.isogeny_class()
Elliptic curves in class 86640.dy
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 86640.dy1 | 86640ec4 | \([0, 1, 0, -305108218440, -64867872366268812]\) | \(16300610738133468173382620881/2228489100\) | \(429429690402394521600\) | \([2]\) | \(207360000\) | \(4.7742\) | |
| 86640.dy2 | 86640ec3 | \([0, 1, 0, -19069261960, -1013565461916940]\) | \(-3979640234041473454886161/1471455901872240\) | \(-283549447193514312677130240\) | \([2]\) | \(103680000\) | \(4.4276\) | |
| 86640.dy3 | 86640ec2 | \([0, 1, 0, -507970440, -3796717459212]\) | \(75224183150104868881/11219310000000000\) | \(2161960235672002560000000000\) | \([2]\) | \(41472000\) | \(3.9695\) | |
| 86640.dy4 | 86640ec1 | \([0, 1, 0, 53918840, -324016953100]\) | \(89962967236397039/287450726400000\) | \(-55391734446399317606400000\) | \([2]\) | \(20736000\) | \(3.6229\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 86640.dy have rank \(1\).
Complex multiplication
The elliptic curves in class 86640.dy do not have complex multiplication.Modular form 86640.2.a.dy
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.
