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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 4730e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4730.h1 | 4730e1 | \([1, 0, 0, -3916, 96400]\) | \(-6641385549974209/198390579200\) | \(-198390579200\) | \([3]\) | \(5760\) | \(0.94599\) | \(\Gamma_0(N)\)-optimal |
4730.h2 | 4730e2 | \([1, 0, 0, 17844, 373136]\) | \(628345970980160831/423295268000000\) | \(-423295268000000\) | \([]\) | \(17280\) | \(1.4953\) |
Rank
sage: E.rank()
The elliptic curves in class 4730e have rank \(1\).
Complex multiplication
The elliptic curves in class 4730e do not have complex multiplication.Modular form 4730.2.a.e
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 Cremona 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 Cremona labels.