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SageMath
E = EllipticCurve("cj1")
E.isogeny_class()
Elliptic curves in class 124215.cj
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 124215.cj1 | 124215bd8 | \([1, 1, 0, -1076530172, 13594808497509]\) | \(242970740812818720001/24375\) | \(13841813018499375\) | \([2]\) | \(24772608\) | \(3.4448\) | |
| 124215.cj2 | 124215bd6 | \([1, 1, 0, -67283297, 212396784384]\) | \(59319456301170001/594140625\) | \(337394192325922265625\) | \([2, 2]\) | \(12386304\) | \(3.0983\) | |
| 124215.cj3 | 124215bd7 | \([1, 1, 0, -65668502, 223078007391]\) | \(-55150149867714721/5950927734375\) | \(-3379348881469573974609375\) | \([2]\) | \(24772608\) | \(3.4448\) | |
| 124215.cj4 | 124215bd4 | \([1, 1, 0, -4306292, 3149387571]\) | \(15551989015681/1445900625\) | \(821082506444364425625\) | \([2, 2]\) | \(6193152\) | \(2.7517\) | |
| 124215.cj5 | 124215bd2 | \([1, 1, 0, -952487, -303019296]\) | \(168288035761/27720225\) | \(15741463437158229225\) | \([2, 2]\) | \(3096576\) | \(2.4051\) | |
| 124215.cj6 | 124215bd1 | \([1, 1, 0, -911082, -335091609]\) | \(147281603041/5265\) | \(2989831611995865\) | \([2]\) | \(1548288\) | \(2.0586\) | \(\Gamma_0(N)\)-optimal |
| 124215.cj7 | 124215bd3 | \([1, 1, 0, 1738838, -1701970031]\) | \(1023887723039/2798036865\) | \(-1588919101710694491465\) | \([2]\) | \(6193152\) | \(2.7517\) | |
| 124215.cj8 | 124215bd5 | \([1, 1, 0, 5009833, 14923106346]\) | \(24487529386319/183539412225\) | \(-104226388740255521601225\) | \([2]\) | \(12386304\) | \(3.0983\) |
Rank
sage: E.rank()
The elliptic curves in class 124215.cj have rank \(1\).
Complex multiplication
The elliptic curves in class 124215.cj do not have complex multiplication.Modular form 124215.2.a.cj
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 16 & 16 & 8 \\ 2 & 1 & 2 & 2 & 4 & 8 & 8 & 4 \\ 4 & 2 & 1 & 4 & 8 & 16 & 16 & 8 \\ 4 & 2 & 4 & 1 & 2 & 4 & 4 & 2 \\ 8 & 4 & 8 & 2 & 1 & 2 & 2 & 4 \\ 16 & 8 & 16 & 4 & 2 & 1 & 4 & 8 \\ 16 & 8 & 16 & 4 & 2 & 4 & 1 & 8 \\ 8 & 4 & 8 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
