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SageMath
E = EllipticCurve("bj1")
E.isogeny_class()
Elliptic curves in class 136242.bj
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
136242.bj1 | 136242e3 | \([1, -1, 1, -905915, 332105131]\) | \(-189613868625/128\) | \(-55504153729152\) | \([]\) | \(1016064\) | \(1.9530\) | |
136242.bj2 | 136242e4 | \([1, -1, 1, -716690, 474568849]\) | \(-1159088625/2097152\) | \(-73659784430572535808\) | \([]\) | \(3048192\) | \(2.5023\) | |
136242.bj3 | 136242e2 | \([1, -1, 1, -35480, -2686877]\) | \(-140625/8\) | \(-280989778253832\) | \([]\) | \(435456\) | \(1.5294\) | |
136242.bj4 | 136242e1 | \([1, -1, 1, 2365, -7451]\) | \(3375/2\) | \(-867252402018\) | \([]\) | \(145152\) | \(0.98006\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 136242.bj have rank \(1\).
Complex multiplication
The elliptic curves in class 136242.bj do not have complex multiplication.Modular form 136242.2.a.bj
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 21 & 7 \\ 3 & 1 & 7 & 21 \\ 21 & 7 & 1 & 3 \\ 7 & 21 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.