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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 271062.x
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 271062.x1 | 271062x4 | \([1, -1, 1, -13314048215, 591310711534331]\) | \(139545621883503188502625/220644468\) | \(412696623801022095348\) | \([2]\) | \(226934784\) | \(4.1125\) | |
| 271062.x2 | 271062x3 | \([1, -1, 1, -832135955, 9239200731299]\) | \(34069730739753390625/1354703543952\) | \(2533857222471899660080272\) | \([2]\) | \(113467392\) | \(3.7659\) | |
| 271062.x3 | 271062x2 | \([1, -1, 1, -164830595, 806398039235]\) | \(264788619837198625/3058196150592\) | \(5720094583430029795029312\) | \([2]\) | \(75644928\) | \(3.5632\) | |
| 271062.x4 | 271062x1 | \([1, -1, 1, -18949955, -11642237629]\) | \(402355893390625/201513996288\) | \(376914711121192936378368\) | \([2]\) | \(37822464\) | \(3.2166\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 271062.x have rank \(1\).
Complex multiplication
The elliptic curves in class 271062.x do not have complex multiplication.Modular form 271062.2.a.x
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
