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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 5525d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
5525.i2 | 5525d1 | \([1, -1, 0, -18317, 957216]\) | \(43499078731809/82055753\) | \(1282121140625\) | \([2]\) | \(9600\) | \(1.2133\) | \(\Gamma_0(N)\)-optimal |
5525.i1 | 5525d2 | \([1, -1, 0, -292942, 61100091]\) | \(177930109857804849/634933\) | \(9920828125\) | \([2]\) | \(19200\) | \(1.5599\) |
Rank
sage: E.rank()
The elliptic curves in class 5525d have rank \(0\).
Complex multiplication
The elliptic curves in class 5525d do not have complex multiplication.Modular form 5525.2.a.d
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.