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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 1785.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1785.a1 | 1785c3 | \([1, 1, 1, -61201, -5853052]\) | \(25351269426118370449/27551475\) | \(27551475\) | \([2]\) | \(3072\) | \(1.1464\) | |
1785.a2 | 1785c4 | \([1, 1, 1, -4771, -44596]\) | \(12010404962647729/6166198828125\) | \(6166198828125\) | \([2]\) | \(3072\) | \(1.1464\) | |
1785.a3 | 1785c2 | \([1, 1, 1, -3826, -92602]\) | \(6193921595708449/6452105625\) | \(6452105625\) | \([2, 2]\) | \(1536\) | \(0.79981\) | |
1785.a4 | 1785c1 | \([1, 1, 1, -181, -2206]\) | \(-656008386769/1581036975\) | \(-1581036975\) | \([2]\) | \(768\) | \(0.45323\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1785.a have rank \(0\).
Complex multiplication
The elliptic curves in class 1785.a do not have complex multiplication.Modular form 1785.2.a.a
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.