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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 414736.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
414736.k1 | 414736k2 | \([0, 1, 0, -1424419432, -20692592772300]\) | \(13062552753151/92\) | \(2251112251258262798336\) | \([2]\) | \(90832896\) | \(3.6953\) | \(\Gamma_0(N)\)-optimal* |
414736.k2 | 414736k1 | \([0, 1, 0, -88969512, -323776412492]\) | \(-3183010111/8464\) | \(-207102327115760177446912\) | \([2]\) | \(45416448\) | \(3.3487\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 414736.k have rank \(0\).
Complex multiplication
The elliptic curves in class 414736.k do not have complex multiplication.Modular form 414736.2.a.k
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.