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SageMath
E = EllipticCurve("cs1")
E.isogeny_class()
Elliptic curves in class 444360cs
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
444360.cs3 | 444360cs1 | \([0, 1, 0, -37735, 2691650]\) | \(2508888064/118125\) | \(279787830210000\) | \([2]\) | \(2365440\) | \(1.5329\) | \(\Gamma_0(N)\)-optimal |
444360.cs2 | 444360cs2 | \([0, 1, 0, -103860, -9396000]\) | \(3269383504/893025\) | \(33843135942201600\) | \([2, 2]\) | \(4730880\) | \(1.8795\) | |
444360.cs4 | 444360cs3 | \([0, 1, 0, 266440, -60941760]\) | \(13799183324/18600435\) | \(-2819616697355996160\) | \([2]\) | \(9461760\) | \(2.2260\) | |
444360.cs1 | 444360cs4 | \([0, 1, 0, -1532160, -730401840]\) | \(2624033547076/324135\) | \(49135219590159360\) | \([2]\) | \(9461760\) | \(2.2260\) |
Rank
sage: E.rank()
The elliptic curves in class 444360cs have rank \(0\).
Complex multiplication
The elliptic curves in class 444360cs do not have complex multiplication.Modular form 444360.2.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.