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SageMath
E = EllipticCurve("fz1")
E.isogeny_class()
Elliptic curves in class 121275fz
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
121275.bd1 | 121275fz1 | \([1, -1, 1, -545, 4232]\) | \(571787/99\) | \(3094331625\) | \([2]\) | \(57344\) | \(0.54121\) | \(\Gamma_0(N)\)-optimal |
121275.bd2 | 121275fz2 | \([1, -1, 1, 1030, 23132]\) | \(3869893/9801\) | \(-306338830875\) | \([2]\) | \(114688\) | \(0.88779\) |
Rank
sage: E.rank()
The elliptic curves in class 121275fz have rank \(2\).
Complex multiplication
The elliptic curves in class 121275fz do not have complex multiplication.Modular form 121275.2.a.fz
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.