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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 1290.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1290.i1 | 1290i2 | \([1, 0, 1, -28, -52]\) | \(2305199161/277350\) | \(277350\) | \([2]\) | \(224\) | \(-0.22485\) | |
1290.i2 | 1290i1 | \([1, 0, 1, 2, -4]\) | \(1685159/7740\) | \(-7740\) | \([2]\) | \(112\) | \(-0.57142\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1290.i have rank \(0\).
Complex multiplication
The elliptic curves in class 1290.i do not have complex multiplication.Modular form 1290.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.