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SageMath
E = EllipticCurve("fr1")
E.isogeny_class()
Elliptic curves in class 458640fr
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
458640.fr3 | 458640fr1 | \([0, 0, 0, -49098, -4177397]\) | \(9538484224/26325\) | \(36124690165200\) | \([2]\) | \(1769472\) | \(1.4744\) | \(\Gamma_0(N)\)-optimal* |
458640.fr2 | 458640fr2 | \([0, 0, 0, -68943, -482258]\) | \(1650587344/950625\) | \(20872043206560000\) | \([2, 2]\) | \(3538944\) | \(1.8210\) | \(\Gamma_0(N)\)-optimal* |
458640.fr1 | 458640fr3 | \([0, 0, 0, -730443, 239377642]\) | \(490757540836/2142075\) | \(188126682768460800\) | \([2]\) | \(7077888\) | \(2.1676\) | \(\Gamma_0(N)\)-optimal* |
458640.fr4 | 458640fr4 | \([0, 0, 0, 275037, -3853262]\) | \(26198797244/15234375\) | \(-1337951487600000000\) | \([2]\) | \(7077888\) | \(2.1676\) |
Rank
sage: E.rank()
The elliptic curves in class 458640fr have rank \(1\).
Complex multiplication
The elliptic curves in class 458640fr do not have complex multiplication.Modular form 458640.2.a.fr
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.