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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 1225j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1225.a2 | 1225j1 | \([0, 1, 1, 82, 424]\) | \(4096/7\) | \(-102942875\) | \([]\) | \(384\) | \(0.22243\) | \(\Gamma_0(N)\)-optimal |
1225.a1 | 1225j2 | \([0, 1, 1, -7268, -242126]\) | \(-2887553024/16807\) | \(-247165842875\) | \([]\) | \(1920\) | \(1.0271\) |
Rank
sage: E.rank()
The elliptic curves in class 1225j have rank \(1\).
Complex multiplication
The elliptic curves in class 1225j do not have complex multiplication.Modular form 1225.2.a.j
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.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.