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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 4002.h
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4002.h1 | 4002k2 | \([1, 1, 1, -92909, -10938229]\) | \(88694637150489389137/6546157250904\) | \(6546157250904\) | \([2]\) | \(16128\) | \(1.5094\) | |
4002.h2 | 4002k1 | \([1, 1, 1, -5429, -195685]\) | \(-17696534894747857/5921086039488\) | \(-5921086039488\) | \([2]\) | \(8064\) | \(1.1628\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 4002.h have rank \(1\).
Complex multiplication
The elliptic curves in class 4002.h do not have complex multiplication.Modular form 4002.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.