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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 3366.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3366.d1 | 3366c3 | \([1, -1, 0, -17433, 825475]\) | \(803760366578833/65593817586\) | \(47817893020194\) | \([2]\) | \(12288\) | \(1.3676\) | |
3366.d2 | 3366c2 | \([1, -1, 0, -3663, -69575]\) | \(7457162887153/1370924676\) | \(999404088804\) | \([2, 2]\) | \(6144\) | \(1.0210\) | |
3366.d3 | 3366c1 | \([1, -1, 0, -3483, -78251]\) | \(6411014266033/296208\) | \(215935632\) | \([2]\) | \(3072\) | \(0.67442\) | \(\Gamma_0(N)\)-optimal |
3366.d4 | 3366c4 | \([1, -1, 0, 7227, -411521]\) | \(57258048889007/132611470002\) | \(-96673761631458\) | \([2]\) | \(12288\) | \(1.3676\) |
Rank
sage: E.rank()
The elliptic curves in class 3366.d have rank \(0\).
Complex multiplication
The elliptic curves in class 3366.d do not have complex multiplication.Modular form 3366.2.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.