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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 218405e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
218405.e2 | 218405e1 | \([0, 1, 1, -4426341, -3584652549]\) | \(318767104/125\) | \(3760927240559625125\) | \([]\) | \(6463800\) | \(2.5289\) | \(\Gamma_0(N)\)-optimal |
218405.e1 | 218405e2 | \([0, 1, 1, -12725731, 13006658000]\) | \(7575076864/1953125\) | \(58764488133744142578125\) | \([]\) | \(19391400\) | \(3.0782\) |
Rank
sage: E.rank()
The elliptic curves in class 218405e have rank \(0\).
Complex multiplication
The elliptic curves in class 218405e do not have complex multiplication.Modular form 218405.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.