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SageMath
E = EllipticCurve("cs1")
E.isogeny_class()
Elliptic curves in class 53040cs
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
53040.db3 | 53040cs1 | \([0, 1, 0, -9400, -353452]\) | \(22428153804601/35802000\) | \(146644992000\) | \([2]\) | \(129024\) | \(1.0403\) | \(\Gamma_0(N)\)-optimal |
53040.db2 | 53040cs2 | \([0, 1, 0, -12280, -121900]\) | \(50002789171321/27473062500\) | \(112529664000000\) | \([2, 2]\) | \(258048\) | \(1.3868\) | |
53040.db4 | 53040cs3 | \([0, 1, 0, 47720, -913900]\) | \(2933972022568679/1789082460750\) | \(-7328081759232000\) | \([2]\) | \(516096\) | \(1.7334\) | |
53040.db1 | 53040cs4 | \([0, 1, 0, -118360, 15535508]\) | \(44769506062996441/323730468750\) | \(1326000000000000\) | \([2]\) | \(516096\) | \(1.7334\) |
Rank
sage: E.rank()
The elliptic curves in class 53040cs have rank \(0\).
Complex multiplication
The elliptic curves in class 53040cs do not have complex multiplication.Modular form 53040.2.a.cs
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.