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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 678e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 678.f3 | 678e1 | \([1, 0, 0, -192, 1008]\) | \(783012621313/146448\) | \(146448\) | \([4]\) | \(160\) | \(-0.0069056\) | \(\Gamma_0(N)\)-optimal |
| 678.f2 | 678e2 | \([1, 0, 0, -212, 780]\) | \(1054045415233/335109636\) | \(335109636\) | \([2, 2]\) | \(320\) | \(0.33967\) | |
| 678.f1 | 678e3 | \([1, 0, 0, -1342, -18430]\) | \(267301555199713/9728558946\) | \(9728558946\) | \([2]\) | \(640\) | \(0.68624\) | |
| 678.f4 | 678e4 | \([1, 0, 0, 598, 5478]\) | \(23647316984927/26413672482\) | \(-26413672482\) | \([2]\) | \(640\) | \(0.68624\) |
Rank
sage: E.rank()
The elliptic curves in class 678e have rank \(0\).
Complex multiplication
The elliptic curves in class 678e do not have complex multiplication.Modular form 678.2.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 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.
