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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 2730.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2730.d1 | 2730f4 | \([1, 1, 0, -7843, 264103]\) | \(53365044437418169/41984670\) | \(41984670\) | \([2]\) | \(3072\) | \(0.76934\) | |
2730.d2 | 2730f3 | \([1, 1, 0, -1143, -9477]\) | \(165369706597369/60703354530\) | \(60703354530\) | \([2]\) | \(3072\) | \(0.76934\) | |
2730.d3 | 2730f2 | \([1, 1, 0, -493, 3913]\) | \(13293525831769/365192100\) | \(365192100\) | \([2, 2]\) | \(1536\) | \(0.42277\) | |
2730.d4 | 2730f1 | \([1, 1, 0, 7, 213]\) | \(30080231/19110000\) | \(-19110000\) | \([2]\) | \(768\) | \(0.076193\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 2730.d have rank \(1\).
Complex multiplication
The elliptic curves in class 2730.d do not have complex multiplication.Modular form 2730.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.