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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 232050x
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 232050.x3 | 232050x1 | \([1, 1, 0, -4550, 52500]\) | \(666940371553/304152576\) | \(4752384000000\) | \([2]\) | \(524288\) | \(1.1275\) | \(\Gamma_0(N)\)-optimal |
| 232050.x2 | 232050x2 | \([1, 1, 0, -36550, -2667500]\) | \(345608484635233/5513953536\) | \(86155524000000\) | \([2, 2]\) | \(1048576\) | \(1.4740\) | |
| 232050.x4 | 232050x3 | \([1, 1, 0, -2550, -7393500]\) | \(-117433042273/1510843540752\) | \(-23606930324250000\) | \([2]\) | \(2097152\) | \(1.8206\) | |
| 232050.x1 | 232050x4 | \([1, 1, 0, -582550, -171381500]\) | \(1399279497274949473/364819728\) | \(5700308250000\) | \([2]\) | \(2097152\) | \(1.8206\) |
Rank
sage: E.rank()
The elliptic curves in class 232050x have rank \(1\).
Complex multiplication
The elliptic curves in class 232050x do not have complex multiplication.Modular form 232050.2.a.x
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.
