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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 123840.z
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
123840.z1 | 123840du1 | \([0, 0, 0, -1188, -6912]\) | \(2299968/1075\) | \(86668185600\) | \([2]\) | \(98304\) | \(0.79318\) | \(\Gamma_0(N)\)-optimal |
123840.z2 | 123840du2 | \([0, 0, 0, 4212, -52272]\) | \(12812904/9245\) | \(-5962771169280\) | \([2]\) | \(196608\) | \(1.1398\) |
Rank
sage: E.rank()
The elliptic curves in class 123840.z have rank \(1\).
Complex multiplication
The elliptic curves in class 123840.z do not have complex multiplication.Modular form 123840.2.a.z
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.