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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 5056i
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 5056.d3 | 5056i1 | \([0, -1, 0, -2977, 63521]\) | \(11134383337/316\) | \(82837504\) | \([]\) | \(2560\) | \(0.62097\) | \(\Gamma_0(N)\)-optimal |
| 5056.d2 | 5056i2 | \([0, -1, 0, -5217, -41759]\) | \(59914169497/31554496\) | \(8271821799424\) | \([]\) | \(7680\) | \(1.1703\) | |
| 5056.d1 | 5056i3 | \([0, -1, 0, -333857, -74137439]\) | \(15698803397448457/20709376\) | \(5428838662144\) | \([]\) | \(23040\) | \(1.7196\) |
Rank
sage: E.rank()
The elliptic curves in class 5056i have rank \(2\).
Complex multiplication
The elliptic curves in class 5056i do not have complex multiplication.Modular form 5056.2.a.i
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
