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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 139656.w
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
139656.w1 | 139656e2 | \([0, 1, 0, -3488, 48720]\) | \(376785500/131769\) | \(1641711025152\) | \([2]\) | \(196608\) | \(1.0453\) | |
139656.w2 | 139656e1 | \([0, 1, 0, 652, 5664]\) | \(9826000/9801\) | \(-30527684352\) | \([2]\) | \(98304\) | \(0.69876\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 139656.w have rank \(2\).
Complex multiplication
The elliptic curves in class 139656.w do not have complex multiplication.Modular form 139656.2.a.w
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.