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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 9633m
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
9633.h3 | 9633m1 | \([1, 0, 0, -257, -1392]\) | \(389017/57\) | \(275128113\) | \([2]\) | \(2880\) | \(0.34432\) | \(\Gamma_0(N)\)-optimal |
9633.h2 | 9633m2 | \([1, 0, 0, -1102, 12635]\) | \(30664297/3249\) | \(15682302441\) | \([2, 2]\) | \(5760\) | \(0.69089\) | |
9633.h1 | 9633m3 | \([1, 0, 0, -17157, 863550]\) | \(115714886617/1539\) | \(7428459051\) | \([2]\) | \(11520\) | \(1.0375\) | |
9633.h4 | 9633m4 | \([1, 0, 0, 1433, 62828]\) | \(67419143/390963\) | \(-1887103727067\) | \([2]\) | \(11520\) | \(1.0375\) |
Rank
sage: E.rank()
The elliptic curves in class 9633m have rank \(1\).
Complex multiplication
The elliptic curves in class 9633m do not have complex multiplication.Modular form 9633.2.a.m
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.