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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 31713.h
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
31713.h1 | 31713e4 | \([1, 0, 1, -140807, -20328541]\) | \(347873904937/395307\) | \(350836417625067\) | \([2]\) | \(172800\) | \(1.7044\) | |
31713.h2 | 31713e2 | \([1, 0, 1, -11072, -141775]\) | \(169112377/88209\) | \(78285812197329\) | \([2, 2]\) | \(86400\) | \(1.3578\) | |
31713.h3 | 31713e1 | \([1, 0, 1, -6267, 188809]\) | \(30664297/297\) | \(263588593257\) | \([2]\) | \(43200\) | \(1.0112\) | \(\Gamma_0(N)\)-optimal |
31713.h4 | 31713e3 | \([1, 0, 1, 41783, -1093165]\) | \(9090072503/5845851\) | \(-5188214281077531\) | \([2]\) | \(172800\) | \(1.7044\) |
Rank
sage: E.rank()
The elliptic curves in class 31713.h have rank \(1\).
Complex multiplication
The elliptic curves in class 31713.h do not have complex multiplication.Modular form 31713.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.