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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 400710h
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
400710.h3 | 400710h1 | \([1, 1, 0, -87008, -9673728]\) | \(1548415333009/43192320\) | \(2032020746833920\) | \([2]\) | \(2903040\) | \(1.7165\) | \(\Gamma_0(N)\)-optimal* |
400710.h2 | 400710h2 | \([1, 1, 0, -202528, 21262528]\) | \(19528130963089/7116609600\) | \(334807168365057600\) | \([2, 2]\) | \(5806080\) | \(2.0631\) | \(\Gamma_0(N)\)-optimal* |
400710.h1 | 400710h3 | \([1, 1, 0, -2873928, 1873611288]\) | \(55799459660732689/15622881480\) | \(734992222985183880\) | \([2]\) | \(11612160\) | \(2.4097\) | \(\Gamma_0(N)\)-optimal* |
400710.h4 | 400710h4 | \([1, 1, 0, 620552, 151144552]\) | \(561740261198831/534135885000\) | \(-25128893283539685000\) | \([2]\) | \(11612160\) | \(2.4097\) |
Rank
sage: E.rank()
The elliptic curves in class 400710h have rank \(1\).
Complex multiplication
The elliptic curves in class 400710h do not have complex multiplication.Modular form 400710.2.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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.