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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 80850w
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 80850.bs6 | 80850w1 | \([1, 1, 0, 312350, -6255500]\) | \(1833318007919/1070530560\) | \(-1967919528960000000\) | \([2]\) | \(1769472\) | \(2.1997\) | \(\Gamma_0(N)\)-optimal |
| 80850.bs5 | 80850w2 | \([1, 1, 0, -1255650, -51727500]\) | \(119102750067601/68309049600\) | \(125570177756100000000\) | \([2, 2]\) | \(3538944\) | \(2.5463\) | |
| 80850.bs3 | 80850w3 | \([1, 1, 0, -13113650, 18197734500]\) | \(135670761487282321/643043610000\) | \(1182084963638906250000\) | \([2, 2]\) | \(7077888\) | \(2.8929\) | |
| 80850.bs2 | 80850w4 | \([1, 1, 0, -14485650, -21180037500]\) | \(182864522286982801/463015182960\) | \(851144894688453750000\) | \([2]\) | \(7077888\) | \(2.8929\) | |
| 80850.bs4 | 80850w5 | \([1, 1, 0, -6376150, 36880822000]\) | \(-15595206456730321/310672490129100\) | \(-571098559237476342187500\) | \([2]\) | \(14155776\) | \(3.2394\) | |
| 80850.bs1 | 80850w6 | \([1, 1, 0, -209579150, 1167717375000]\) | \(553808571467029327441/12529687500\) | \(23032893823242187500\) | \([2]\) | \(14155776\) | \(3.2394\) |
Rank
sage: E.rank()
The elliptic curves in class 80850w have rank \(1\).
Complex multiplication
The elliptic curves in class 80850w do not have complex multiplication.Modular form 80850.2.a.w
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
