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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 5040.r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
5040.r1 | 5040bh1 | \([0, 0, 0, -48, 47]\) | \(1048576/525\) | \(6123600\) | \([2]\) | \(768\) | \(-0.0049075\) | \(\Gamma_0(N)\)-optimal |
5040.r2 | 5040bh2 | \([0, 0, 0, 177, 362]\) | \(3286064/2205\) | \(-411505920\) | \([2]\) | \(1536\) | \(0.34167\) |
Rank
sage: E.rank()
The elliptic curves in class 5040.r have rank \(0\).
Complex multiplication
The elliptic curves in class 5040.r do not have complex multiplication.Modular form 5040.2.a.r
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 LMFDB 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 LMFDB labels.