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SageMath
E = EllipticCurve("gd1")
E.isogeny_class()
Elliptic curves in class 73920gd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 73920.er6 | 73920gd1 | \([0, 1, 0, 2239, 11843775]\) | \(4733169839/231139696095\) | \(-60591884493127680\) | \([2]\) | \(491520\) | \(1.8991\) | \(\Gamma_0(N)\)-optimal |
| 73920.er5 | 73920gd2 | \([0, 1, 0, -766081, 253249919]\) | \(189674274234120481/3859869269025\) | \(1011841569659289600\) | \([2, 2]\) | \(983040\) | \(2.2457\) | |
| 73920.er4 | 73920gd3 | \([0, 1, 0, -1628481, -422354241]\) | \(1821931919215868881/761147600816295\) | \(199530276668386836480\) | \([2]\) | \(1966080\) | \(2.5923\) | |
| 73920.er2 | 73920gd4 | \([0, 1, 0, -12196801, 16391140415]\) | \(765458482133960722801/326869475625\) | \(85686871818240000\) | \([2, 2]\) | \(1966080\) | \(2.5923\) | |
| 73920.er3 | 73920gd5 | \([0, 1, 0, -12136321, 16561802879]\) | \(-754127868744065783521/15825714261328125\) | \(-4148616039321600000000\) | \([2]\) | \(3932160\) | \(2.9389\) | |
| 73920.er1 | 73920gd6 | \([0, 1, 0, -195148801, 1049228361215]\) | \(3135316978843283198764801/571725\) | \(149874278400\) | \([2]\) | \(3932160\) | \(2.9389\) |
Rank
sage: E.rank()
The elliptic curves in class 73920gd have rank \(1\).
Complex multiplication
The elliptic curves in class 73920gd do not have complex multiplication.Modular form 73920.2.a.gd
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.
