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SageMath
E = EllipticCurve("bb1")
E.isogeny_class()
Elliptic curves in class 61710bb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 61710.t3 | 61710bb1 | \([1, 0, 1, -10500504, -13097617898]\) | \(72276643492008825169/66646800\) | \(118068871654800\) | \([2]\) | \(2457600\) | \(2.4297\) | \(\Gamma_0(N)\)-optimal |
| 61710.t2 | 61710bb2 | \([1, 0, 1, -10502924, -13091279434]\) | \(72326626749631816849/69403061722500\) | \(122951757428173822500\) | \([2, 2]\) | \(4915200\) | \(2.7762\) | |
| 61710.t4 | 61710bb3 | \([1, 0, 1, -8052674, -19358038834]\) | \(-32597768919523300849/72509045805004050\) | \(-128454197695358779822050\) | \([2]\) | \(9830400\) | \(3.1228\) | |
| 61710.t1 | 61710bb4 | \([1, 0, 1, -12991894, -6418848658]\) | \(136894171818794254129/69177425857031250\) | \(122552029728708138281250\) | \([2]\) | \(9830400\) | \(3.1228\) |
Rank
sage: E.rank()
The elliptic curves in class 61710bb have rank \(1\).
Complex multiplication
The elliptic curves in class 61710bb do not have complex multiplication.Modular form 61710.2.a.bb
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.
