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SageMath
E = EllipticCurve("ft1")
E.isogeny_class()
Elliptic curves in class 411840.ft
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
411840.ft1 | 411840ft3 | \([0, 0, 0, -21973548, 39645889072]\) | \(6139836723518159689/3799803150\) | \(726153690179174400\) | \([2]\) | \(18874368\) | \(2.7481\) | \(\Gamma_0(N)\)-optimal* |
411840.ft2 | 411840ft4 | \([0, 0, 0, -3092268, -1203722192]\) | \(17111482619973769/6627044531250\) | \(1266447932006400000000\) | \([2]\) | \(18874368\) | \(2.7481\) | |
411840.ft3 | 411840ft2 | \([0, 0, 0, -1381548, 611693872]\) | \(1525998818291689/37268302500\) | \(7122083518218240000\) | \([2, 2]\) | \(9437184\) | \(2.4016\) | \(\Gamma_0(N)\)-optimal* |
411840.ft4 | 411840ft1 | \([0, 0, 0, 12372, 30150448]\) | \(1095912791/2055596400\) | \(-392830589494886400\) | \([2]\) | \(4718592\) | \(2.0550\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 411840.ft have rank \(0\).
Complex multiplication
The elliptic curves in class 411840.ft do not have complex multiplication.Modular form 411840.2.a.ft
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.