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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 124950ca
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
124950.i2 | 124950ca1 | \([1, 1, 0, -22075, 2619625]\) | \(-5177717/9996\) | \(-2296912898437500\) | \([2]\) | \(890880\) | \(1.6372\) | \(\Gamma_0(N)\)-optimal |
124950.i1 | 124950ca2 | \([1, 1, 0, -450825, 116238375]\) | \(44099220437/36414\) | \(8367325558593750\) | \([2]\) | \(1781760\) | \(1.9838\) |
Rank
sage: E.rank()
The elliptic curves in class 124950ca have rank \(1\).
Complex multiplication
The elliptic curves in class 124950ca do not have complex multiplication.Modular form 124950.2.a.ca
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.