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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 486720ca
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
486720.ca2 | 486720ca1 | \([0, 0, 0, 4378452, 3730013872]\) | \(40254822716/49359375\) | \(-11382488617982976000000\) | \([2]\) | \(20643840\) | \(2.9183\) | \(\Gamma_0(N)\)-optimal* |
486720.ca1 | 486720ca2 | \([0, 0, 0, -26041548, 35841365872]\) | \(4234737878642/1247410125\) | \(575316504707331538944000\) | \([2]\) | \(41287680\) | \(3.2649\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 486720ca have rank \(0\).
Complex multiplication
The elliptic curves in class 486720ca do not have complex multiplication.Modular form 486720.2.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.