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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 1170.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1170.e1 | 1170e2 | \([1, -1, 0, -414, 3010]\) | \(10779215329/1232010\) | \(898135290\) | \([2]\) | \(768\) | \(0.45026\) | |
1170.e2 | 1170e1 | \([1, -1, 0, 36, 220]\) | \(6967871/35100\) | \(-25587900\) | \([2]\) | \(384\) | \(0.10369\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1170.e have rank \(1\).
Complex multiplication
The elliptic curves in class 1170.e do not have complex multiplication.Modular form 1170.2.a.e
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.