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SageMath
E = EllipticCurve("ir1")
E.isogeny_class()
Elliptic curves in class 485100.ir
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
485100.ir1 | 485100ir1 | \([0, 0, 0, -58800, -5016375]\) | \(28311552/2695\) | \(2140182371250000\) | \([2]\) | \(3096576\) | \(1.6795\) | \(\Gamma_0(N)\)-optimal |
485100.ir2 | 485100ir2 | \([0, 0, 0, 69825, -23924250]\) | \(2963088/21175\) | \(-269051498100000000\) | \([2]\) | \(6193152\) | \(2.0261\) |
Rank
sage: E.rank()
The elliptic curves in class 485100.ir have rank \(0\).
Complex multiplication
The elliptic curves in class 485100.ir do not have complex multiplication.Modular form 485100.2.a.ir
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.