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SageMath
E = EllipticCurve("y1")
E.isogeny_class()
Elliptic curves in class 38115.y
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
38115.y1 | 38115z4 | \([1, -1, 0, -2312514, -1338336455]\) | \(1058993490188089/13182390375\) | \(17024634924166398375\) | \([2]\) | \(1105920\) | \(2.4999\) | |
38115.y2 | 38115z2 | \([1, -1, 0, -270639, 21143920]\) | \(1697509118089/833765625\) | \(1076781598340765625\) | \([2, 2]\) | \(552960\) | \(2.1533\) | |
38115.y3 | 38115z1 | \([1, -1, 0, -221634, 40187263]\) | \(932288503609/779625\) | \(1006860715331625\) | \([2]\) | \(276480\) | \(1.8067\) | \(\Gamma_0(N)\)-optimal |
38115.y4 | 38115z3 | \([1, -1, 0, 987156, 161262283]\) | \(82375335041831/56396484375\) | \(-72834253134521484375\) | \([2]\) | \(1105920\) | \(2.4999\) |
Rank
sage: E.rank()
The elliptic curves in class 38115.y have rank \(0\).
Complex multiplication
The elliptic curves in class 38115.y do not have complex multiplication.Modular form 38115.2.a.y
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.