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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 52a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
52.a2 | 52a1 | \([0, 0, 0, 1, -10]\) | \(432/169\) | \(-43264\) | \([2]\) | \(3\) | \(-0.43130\) | \(\Gamma_0(N)\)-optimal |
52.a1 | 52a2 | \([0, 0, 0, -4, -3]\) | \(442368/13\) | \(208\) | \([2]\) | \(6\) | \(-0.77787\) |
Rank
sage: E.rank()
The elliptic curves in class 52a have rank \(0\).
Complex multiplication
The elliptic curves in class 52a do not have complex multiplication.Modular form 52.2.a.a
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.