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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 10010.c
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 10010.c1 | 10010a3 | \([1, -1, 0, -822155, -286726349]\) | \(61458947171027474307849/96346250\) | \(96346250\) | \([2]\) | \(57344\) | \(1.6893\) | |
| 10010.c2 | 10010a2 | \([1, -1, 0, -51385, -4470375]\) | \(15005053520986088169/594086392900\) | \(594086392900\) | \([2, 2]\) | \(28672\) | \(1.3427\) | |
| 10010.c3 | 10010a4 | \([1, -1, 0, -48935, -4917745]\) | \(-12959477208091719369/2999928960118090\) | \(-2999928960118090\) | \([2]\) | \(57344\) | \(1.6893\) | |
| 10010.c4 | 10010a1 | \([1, -1, 0, -3365, -62139]\) | \(4214552938238889/725442557840\) | \(725442557840\) | \([2]\) | \(14336\) | \(0.99615\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 10010.c have rank \(1\).
Complex multiplication
The elliptic curves in class 10010.c do not have complex multiplication.Modular form 10010.2.a.c
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
