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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 212160.w
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
212160.w1 | 212160hk1 | \([0, -1, 0, -1468041, -684103095]\) | \(85423746487501618624/5335675338825\) | \(21854926187827200\) | \([2]\) | \(3612672\) | \(2.1944\) | \(\Gamma_0(N)\)-optimal |
212160.w2 | 212160hk2 | \([0, -1, 0, -1380161, -769680639]\) | \(-8872854201096612488/2684655854083125\) | \(-87970803026595840000\) | \([2]\) | \(7225344\) | \(2.5409\) |
Rank
sage: E.rank()
The elliptic curves in class 212160.w have rank \(1\).
Complex multiplication
The elliptic curves in class 212160.w do not have complex multiplication.Modular form 212160.2.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.