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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 272d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
272.d4 | 272d1 | \([0, -1, 0, -48, -64]\) | \(3048625/1088\) | \(4456448\) | \([2]\) | \(48\) | \(-0.023245\) | \(\Gamma_0(N)\)-optimal |
272.d3 | 272d2 | \([0, -1, 0, -688, -6720]\) | \(8805624625/2312\) | \(9469952\) | \([2]\) | \(96\) | \(0.32333\) | |
272.d2 | 272d3 | \([0, -1, 0, -1648, 26304]\) | \(120920208625/19652\) | \(80494592\) | \([2]\) | \(144\) | \(0.52606\) | |
272.d1 | 272d4 | \([0, -1, 0, -1808, 21056]\) | \(159661140625/48275138\) | \(197734965248\) | \([2]\) | \(288\) | \(0.87263\) |
Rank
sage: E.rank()
The elliptic curves in class 272d have rank \(0\).
Complex multiplication
The elliptic curves in class 272d do not have complex multiplication.Modular form 272.2.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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.