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SageMath
E = EllipticCurve("lw1")
E.isogeny_class()
Elliptic curves in class 100800lw
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 100800.gr4 | 100800lw1 | \([0, 0, 0, -157800, 24127000]\) | \(37256083456/525\) | \(6123600000000\) | \([2]\) | \(393216\) | \(1.5938\) | \(\Gamma_0(N)\)-optimal |
| 100800.gr3 | 100800lw2 | \([0, 0, 0, -162300, 22678000]\) | \(2533446736/275625\) | \(51438240000000000\) | \([2, 2]\) | \(786432\) | \(1.9404\) | |
| 100800.gr5 | 100800lw3 | \([0, 0, 0, 215700, 112642000]\) | \(1486779836/8203125\) | \(-6123600000000000000\) | \([2]\) | \(1572864\) | \(2.2869\) | |
| 100800.gr2 | 100800lw4 | \([0, 0, 0, -612300, -160022000]\) | \(34008619684/4862025\) | \(3629482214400000000\) | \([2, 2]\) | \(1572864\) | \(2.2869\) | |
| 100800.gr6 | 100800lw5 | \([0, 0, 0, 1007700, -863102000]\) | \(75798394558/259416045\) | \(-387306079856640000000\) | \([2]\) | \(3145728\) | \(2.6335\) | |
| 100800.gr1 | 100800lw6 | \([0, 0, 0, -9432300, -11149742000]\) | \(62161150998242/1607445\) | \(2399902525440000000\) | \([2]\) | \(3145728\) | \(2.6335\) |
Rank
sage: E.rank()
The elliptic curves in class 100800lw have rank \(1\).
Complex multiplication
The elliptic curves in class 100800lw do not have complex multiplication.Modular form 100800.2.a.lw
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.
