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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 51376.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
51376.i1 | 51376v3 | \([0, -1, 0, -231248, 344373184]\) | \(-69173457625/2550136832\) | \(-50417759895261544448\) | \([]\) | \(886464\) | \(2.4608\) | |
51376.i2 | 51376v1 | \([0, -1, 0, -41968, -3296320]\) | \(-413493625/152\) | \(-3005132668928\) | \([]\) | \(98496\) | \(1.3622\) | \(\Gamma_0(N)\)-optimal |
51376.i3 | 51376v2 | \([0, -1, 0, 25632, -12587264]\) | \(94196375/3511808\) | \(-69430585182912512\) | \([]\) | \(295488\) | \(1.9115\) |
Rank
sage: E.rank()
The elliptic curves in class 51376.i have rank \(1\).
Complex multiplication
The elliptic curves in class 51376.i do not have complex multiplication.Modular form 51376.2.a.i
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.
\(\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.