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SageMath
E = EllipticCurve("du1")
E.isogeny_class()
Elliptic curves in class 111600du
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 111600.fq4 | 111600du1 | \([0, 0, 0, 5000325, -4828625750]\) | \(296354077829711/387386634240\) | \(-18073910807101440000000\) | \([2]\) | \(6635520\) | \(2.9566\) | \(\Gamma_0(N)\)-optimal |
| 111600.fq3 | 111600du2 | \([0, 0, 0, -30711675, -47075921750]\) | \(68663623745397169/19216056254400\) | \(896544320605286400000000\) | \([2]\) | \(13271040\) | \(3.3032\) | |
| 111600.fq2 | 111600du3 | \([0, 0, 0, -142743675, -660353201750]\) | \(-6894246873502147249/47925198774000\) | \(-2235998073999744000000000\) | \([2]\) | \(19906560\) | \(3.5059\) | |
| 111600.fq1 | 111600du4 | \([0, 0, 0, -2287695675, -42115840505750]\) | \(28379906689597370652529/1357352437500\) | \(63328635324000000000000\) | \([2]\) | \(39813120\) | \(3.8525\) |
Rank
sage: E.rank()
The elliptic curves in class 111600du have rank \(1\).
Complex multiplication
The elliptic curves in class 111600du do not have complex multiplication.Modular form 111600.2.a.du
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
