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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 784.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
784.e1 | 784c4 | \([0, 0, 0, -14651, 682570]\) | \(1443468546/7\) | \(1686616064\) | \([4]\) | \(768\) | \(0.97092\) | |
784.e2 | 784c3 | \([0, 0, 0, -2891, -47334]\) | \(11090466/2401\) | \(578509309952\) | \([2]\) | \(768\) | \(0.97092\) | |
784.e3 | 784c2 | \([0, 0, 0, -931, 10290]\) | \(740772/49\) | \(5903156224\) | \([2, 2]\) | \(384\) | \(0.62435\) | |
784.e4 | 784c1 | \([0, 0, 0, 49, 686]\) | \(432/7\) | \(-210827008\) | \([2]\) | \(192\) | \(0.27778\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 784.e have rank \(0\).
Complex multiplication
The elliptic curves in class 784.e do not have complex multiplication.Modular form 784.2.a.e
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.