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SageMath
E = EllipticCurve("by1")
E.isogeny_class()
Elliptic curves in class 69825by
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
69825.q3 | 69825by1 | \([1, 0, 0, -1863, 26592]\) | \(389017/57\) | \(104781140625\) | \([2]\) | \(73728\) | \(0.83952\) | \(\Gamma_0(N)\)-optimal |
69825.q2 | 69825by2 | \([1, 0, 0, -7988, -249033]\) | \(30664297/3249\) | \(5972525015625\) | \([2, 2]\) | \(147456\) | \(1.1861\) | |
69825.q4 | 69825by3 | \([1, 0, 0, 10387, -1222908]\) | \(67419143/390963\) | \(-718693843546875\) | \([2]\) | \(294912\) | \(1.5327\) | |
69825.q1 | 69825by4 | \([1, 0, 0, -124363, -16890658]\) | \(115714886617/1539\) | \(2829090796875\) | \([2]\) | \(294912\) | \(1.5327\) |
Rank
sage: E.rank()
The elliptic curves in class 69825by have rank \(0\).
Complex multiplication
The elliptic curves in class 69825by do not have complex multiplication.Modular form 69825.2.a.by
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.