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SageMath
E = EllipticCurve("t1")
E.isogeny_class()
Elliptic curves in class 5586.t
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
5586.t1 | 5586w4 | \([1, 1, 1, -59634, 5580315]\) | \(199350693197713/547428\) | \(64404356772\) | \([2]\) | \(12288\) | \(1.3072\) | |
5586.t2 | 5586w3 | \([1, 1, 1, -10634, -316933]\) | \(1130389181713/295568028\) | \(34773282926172\) | \([2]\) | \(12288\) | \(1.3072\) | |
5586.t3 | 5586w2 | \([1, 1, 1, -3774, 83691]\) | \(50529889873/2547216\) | \(299677415184\) | \([2, 2]\) | \(6144\) | \(0.96064\) | |
5586.t4 | 5586w1 | \([1, 1, 1, 146, 5291]\) | \(2924207/102144\) | \(-12017139456\) | \([4]\) | \(3072\) | \(0.61407\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 5586.t have rank \(1\).
Complex multiplication
The elliptic curves in class 5586.t do not have complex multiplication.Modular form 5586.2.a.t
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.