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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 349830.z
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
349830.z1 | 349830z2 | \([1, -1, 0, -704813940, 7202287714896]\) | \(65113766972032185121/7786880\) | \(4630605985583809920\) | \([3]\) | \(56609280\) | \(3.4457\) | |
349830.z2 | 349830z1 | \([1, -1, 0, -8804340, 9635925456]\) | \(126922848287521/6029312000\) | \(3585437073147691008000\) | \([]\) | \(18869760\) | \(2.8964\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 349830.z have rank \(1\).
Complex multiplication
The elliptic curves in class 349830.z do not have complex multiplication.Modular form 349830.2.a.z
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.