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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 1710.b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1710.b1 | 1710c4 | \([1, -1, 0, -475411005, 3989926514025]\) | \(16300610738133468173382620881/2228489100\) | \(1624568553900\) | \([2]\) | \(192000\) | \(3.1581\) | |
| 1710.b2 | 1710c3 | \([1, -1, 0, -29713185, 62348184621]\) | \(-3979640234041473454886161/1471455901872240\) | \(-1072691352464862960\) | \([2]\) | \(96000\) | \(2.8116\) | |
| 1710.b3 | 1710c2 | \([1, -1, 0, -791505, 233711325]\) | \(75224183150104868881/11219310000000000\) | \(8178876990000000000\) | \([2]\) | \(38400\) | \(2.3534\) | |
| 1710.b4 | 1710c1 | \([1, -1, 0, 84015, 19909341]\) | \(89962967236397039/287450726400000\) | \(-209551579545600000\) | \([2]\) | \(19200\) | \(2.0068\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1710.b have rank \(0\).
Complex multiplication
The elliptic curves in class 1710.b do not have complex multiplication.Modular form 1710.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
