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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 6699.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6699.d1 | 6699b3 | \([1, 1, 0, -1249, -17510]\) | \(215751695207833/163381911\) | \(163381911\) | \([2]\) | \(3328\) | \(0.50861\) | |
6699.d2 | 6699b2 | \([1, 1, 0, -94, -185]\) | \(93391282153/44876601\) | \(44876601\) | \([2, 2]\) | \(1664\) | \(0.16203\) | |
6699.d3 | 6699b1 | \([1, 1, 0, -49, 112]\) | \(13430356633/180873\) | \(180873\) | \([2]\) | \(832\) | \(-0.18454\) | \(\Gamma_0(N)\)-optimal |
6699.d4 | 6699b4 | \([1, 1, 0, 341, -968]\) | \(4365111505607/3058314567\) | \(-3058314567\) | \([2]\) | \(3328\) | \(0.50861\) |
Rank
sage: E.rank()
The elliptic curves in class 6699.d have rank \(0\).
Complex multiplication
The elliptic curves in class 6699.d do not have complex multiplication.Modular form 6699.2.a.d
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.