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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 637b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 637.b2 | 637b1 | \([0, -1, 1, -359, -2507]\) | \(-43614208/91\) | \(-10706059\) | \([]\) | \(192\) | \(0.23445\) | \(\Gamma_0(N)\)-optimal |
| 637.b3 | 637b2 | \([0, -1, 1, 621, -13238]\) | \(224755712/753571\) | \(-88656874579\) | \([]\) | \(576\) | \(0.78375\) | |
| 637.b1 | 637b3 | \([0, -1, 1, -5749, 415463]\) | \(-178643795968/524596891\) | \(-61718299629259\) | \([]\) | \(1728\) | \(1.3331\) |
Rank
sage: E.rank()
The elliptic curves in class 637b have rank \(0\).
Complex multiplication
The elliptic curves in class 637b do not have complex multiplication.Modular form 637.2.a.b
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
