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SageMath
E = EllipticCurve("gn1")
E.isogeny_class()
Elliptic curves in class 405720gn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 405720.gn4 | 405720gn1 | \([0, 0, 0, -449967, -7519246]\) | \(458891455696/264449745\) | \(5806292179990821120\) | \([4]\) | \(5898240\) | \(2.2900\) | \(\Gamma_0(N)\)-optimal |
| 405720.gn2 | 405720gn2 | \([0, 0, 0, -5115747, -4442809714]\) | \(168591300897604/472410225\) | \(41489195539442918400\) | \([2, 2]\) | \(11796480\) | \(2.6366\) | |
| 405720.gn3 | 405720gn3 | \([0, 0, 0, -3087147, -8001379834]\) | \(-18524646126002/146738831715\) | \(-25774531371555486750720\) | \([2]\) | \(23592960\) | \(2.9832\) | |
| 405720.gn1 | 405720gn4 | \([0, 0, 0, -81796827, -284742829546]\) | \(344577854816148242/2716875\) | \(477216419823360000\) | \([2]\) | \(23592960\) | \(2.9832\) |
Rank
sage: E.rank()
The elliptic curves in class 405720gn have rank \(0\).
Complex multiplication
The elliptic curves in class 405720gn do not have complex multiplication.Modular form 405720.2.a.gn
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
