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SageMath
E = EllipticCurve("cu1")
E.isogeny_class()
Elliptic curves in class 126350cu
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 126350.dh4 | 126350cu1 | \([1, -1, 1, 20870, 1843497]\) | \(1367631/2800\) | \(-2058257293750000\) | \([2]\) | \(691200\) | \(1.6228\) | \(\Gamma_0(N)\)-optimal |
| 126350.dh3 | 126350cu2 | \([1, -1, 1, -159630, 19893497]\) | \(611960049/122500\) | \(90048756601562500\) | \([2, 2]\) | \(1382400\) | \(1.9694\) | |
| 126350.dh2 | 126350cu3 | \([1, -1, 1, -791380, -253022503]\) | \(74565301329/5468750\) | \(4020033776855468750\) | \([2]\) | \(2764800\) | \(2.3159\) | |
| 126350.dh1 | 126350cu4 | \([1, -1, 1, -2415880, 1445843497]\) | \(2121328796049/120050\) | \(88247781469531250\) | \([2]\) | \(2764800\) | \(2.3159\) |
Rank
sage: E.rank()
The elliptic curves in class 126350cu have rank \(0\).
Complex multiplication
The elliptic curves in class 126350cu do not have complex multiplication.Modular form 126350.2.a.cu
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.
