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SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 212415.u
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
212415.u1 | 212415r2 | \([1, 0, 0, -200565302071, 32108194182955526]\) | \(63953244990201015504593/5088175635498046875\) | \(70988929398889472991835658935546875\) | \([2]\) | \(1654456320\) | \(5.4308\) | |
212415.u2 | 212415r1 | \([1, 0, 0, 12663884624, 2265106317334055]\) | \(16098893047132187167/168182866341984375\) | \(-2346444478362695924820470945484375\) | \([2]\) | \(827228160\) | \(5.0843\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 212415.u have rank \(1\).
Complex multiplication
The elliptic curves in class 212415.u do not have complex multiplication.Modular form 212415.2.a.u
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.