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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 1680d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1680.h5 | 1680d1 | \([0, -1, 0, -15, -90]\) | \(-24918016/229635\) | \(-3674160\) | \([2]\) | \(256\) | \(-0.057811\) | \(\Gamma_0(N)\)-optimal |
| 1680.h4 | 1680d2 | \([0, -1, 0, -420, -3168]\) | \(32082281296/99225\) | \(25401600\) | \([2, 2]\) | \(512\) | \(0.28876\) | |
| 1680.h1 | 1680d3 | \([0, -1, 0, -6720, -209808]\) | \(32779037733124/315\) | \(322560\) | \([2]\) | \(1024\) | \(0.63534\) | |
| 1680.h3 | 1680d4 | \([0, -1, 0, -600, 0]\) | \(23366901604/13505625\) | \(13829760000\) | \([2, 4]\) | \(1024\) | \(0.63534\) | |
| 1680.h2 | 1680d5 | \([0, -1, 0, -6480, 202272]\) | \(14695548366242/57421875\) | \(117600000000\) | \([4]\) | \(2048\) | \(0.98191\) | |
| 1680.h6 | 1680d6 | \([0, -1, 0, 2400, -2400]\) | \(746185003198/432360075\) | \(-885473433600\) | \([4]\) | \(2048\) | \(0.98191\) |
Rank
sage: E.rank()
The elliptic curves in class 1680d have rank \(1\).
Complex multiplication
The elliptic curves in class 1680d do not have complex multiplication.Modular form 1680.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
