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SageMath
E = EllipticCurve("el1")
E.isogeny_class()
Elliptic curves in class 84150el
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 84150.hb3 | 84150el1 | \([1, -1, 1, -29105, 1514397]\) | \(6462919457883/1414187500\) | \(596610351562500\) | \([2]\) | \(387072\) | \(1.5488\) | \(\Gamma_0(N)\)-optimal |
| 84150.hb4 | 84150el2 | \([1, -1, 1, 64645, 9201897]\) | \(70819203762117/127995282250\) | \(-53998009699218750\) | \([2]\) | \(774144\) | \(1.8954\) | |
| 84150.hb1 | 84150el3 | \([1, -1, 1, -750980, -250179353]\) | \(152298969481827/86468800\) | \(26593209225000000\) | \([2]\) | \(1161216\) | \(2.0982\) | |
| 84150.hb2 | 84150el4 | \([1, -1, 1, -615980, -343059353]\) | \(-84044939142627/116825833960\) | \(-35929420153666875000\) | \([2]\) | \(2322432\) | \(2.4447\) |
Rank
sage: E.rank()
The elliptic curves in class 84150el have rank \(0\).
Complex multiplication
The elliptic curves in class 84150el do not have complex multiplication.Modular form 84150.2.a.el
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
