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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 55470.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
55470.i1 | 55470g2 | \([1, 0, 1, -3291259, -2689109818]\) | \(-337335507529/72000000\) | \(-841550419987272000000\) | \([]\) | \(3120768\) | \(2.7364\) | |
55470.i2 | 55470g1 | \([1, 0, 1, 286556, 21442826]\) | \(222641831/145800\) | \(-1704139600474225800\) | \([3]\) | \(1040256\) | \(2.1871\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 55470.i have rank \(0\).
Complex multiplication
The elliptic curves in class 55470.i do not have complex multiplication.Modular form 55470.2.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 LMFDB numbering.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.