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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 304920.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
304920.i1 | 304920i4 | \([0, 0, 0, -17119443, -27262955522]\) | \(419574424137124/10761135\) | \(14231205030486850560\) | \([2]\) | \(11796480\) | \(2.7823\) | |
304920.i2 | 304920i3 | \([0, 0, 0, -4704843, 3537928438]\) | \(8709145038724/951192165\) | \(1257916634584333194240\) | \([2]\) | \(11796480\) | \(2.7823\) | |
304920.i3 | 304920i2 | \([0, 0, 0, -1111143, -391423142]\) | \(458891455696/65367225\) | \(21611437391338502400\) | \([2, 2]\) | \(5898240\) | \(2.4357\) | |
304920.i4 | 304920i1 | \([0, 0, 0, 113982, -32951567]\) | \(7925540864/27286875\) | \(-563842000585710000\) | \([2]\) | \(2949120\) | \(2.0891\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 304920.i have rank \(1\).
Complex multiplication
The elliptic curves in class 304920.i do not have complex multiplication.Modular form 304920.2.a.i
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.