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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 33930e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
33930.l1 | 33930e1 | \([1, -1, 0, -11409, 471915]\) | \(-6083088015781323/11781250\) | \(-318093750\) | \([3]\) | \(44928\) | \(0.88498\) | \(\Gamma_0(N)\)-optimal |
33930.l2 | 33930e2 | \([1, -1, 0, -8034, 753740]\) | \(-2913790403187/10716526600\) | \(-210933393067800\) | \([]\) | \(134784\) | \(1.4343\) |
Rank
sage: E.rank()
The elliptic curves in class 33930e have rank \(1\).
Complex multiplication
The elliptic curves in class 33930e do not have complex multiplication.Modular form 33930.2.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.