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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 1785.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1785.e1 | 1785o5 | \([1, 0, 0, -71740, -7387525]\) | \(40832710302042509761/91556816413125\) | \(91556816413125\) | \([2]\) | \(8192\) | \(1.5612\) | |
1785.e2 | 1785o3 | \([1, 0, 0, -6115, -24400]\) | \(25288177725059761/14387797265625\) | \(14387797265625\) | \([2, 2]\) | \(4096\) | \(1.2146\) | |
1785.e3 | 1785o2 | \([1, 0, 0, -3910, 93347]\) | \(6610905152742241/35128130625\) | \(35128130625\) | \([2, 4]\) | \(2048\) | \(0.86806\) | |
1785.e4 | 1785o1 | \([1, 0, 0, -3905, 93600]\) | \(6585576176607121/187425\) | \(187425\) | \([4]\) | \(1024\) | \(0.52149\) | \(\Gamma_0(N)\)-optimal |
1785.e5 | 1785o4 | \([1, 0, 0, -1785, 194922]\) | \(-629004249876241/16074715228425\) | \(-16074715228425\) | \([8]\) | \(4096\) | \(1.2146\) | |
1785.e6 | 1785o6 | \([1, 0, 0, 24230, -188263]\) | \(1573196002879828319/926055908203125\) | \(-926055908203125\) | \([2]\) | \(8192\) | \(1.5612\) |
Rank
sage: E.rank()
The elliptic curves in class 1785.e have rank \(1\).
Complex multiplication
The elliptic curves in class 1785.e do not have complex multiplication.Modular form 1785.2.a.e
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.