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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 5225.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
5225.a1 | 5225b3 | \([1, -1, 1, -25419730, -43387285978]\) | \(116256292809537371612841/15216540068579856875\) | \(237758438571560263671875\) | \([2]\) | \(423936\) | \(3.2137\) | |
5225.a2 | 5225b2 | \([1, -1, 1, -24560355, -46841973478]\) | \(104859453317683374662841/2223652969140625\) | \(34744577642822265625\) | \([2, 2]\) | \(211968\) | \(2.8671\) | |
5225.a3 | 5225b1 | \([1, -1, 1, -24560230, -46842474228]\) | \(104857852278310619039721/47155625\) | \(736806640625\) | \([2]\) | \(105984\) | \(2.5205\) | \(\Gamma_0(N)\)-optimal |
5225.a4 | 5225b4 | \([1, -1, 1, -23702980, -50264614478]\) | \(-94256762600623910012361/15323275604248046875\) | \(-239426181316375732421875\) | \([2]\) | \(423936\) | \(3.2137\) |
Rank
sage: E.rank()
The elliptic curves in class 5225.a have rank \(0\).
Complex multiplication
The elliptic curves in class 5225.a do not have complex multiplication.Modular form 5225.2.a.a
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.