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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 5040.h
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
5040.h1 | 5040bf2 | \([0, 0, 0, -7248, -237508]\) | \(-225637236736/1715\) | \(-320060160\) | \([]\) | \(4320\) | \(0.80685\) | |
5040.h2 | 5040bf1 | \([0, 0, 0, -48, -628]\) | \(-65536/875\) | \(-163296000\) | \([]\) | \(1440\) | \(0.25755\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 5040.h have rank \(1\).
Complex multiplication
The elliptic curves in class 5040.h do not have complex multiplication.Modular form 5040.2.a.h
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.