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SageMath
E = EllipticCurve("gd1")
E.isogeny_class()
Elliptic curves in class 28224gd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 28224.by4 | 28224gd1 | \([0, 0, 0, 1764, -148176]\) | \(432/7\) | \(-9836344885248\) | \([2]\) | \(49152\) | \(1.1737\) | \(\Gamma_0(N)\)-optimal |
| 28224.by3 | 28224gd2 | \([0, 0, 0, -33516, -2222640]\) | \(740772/49\) | \(275417656786944\) | \([2, 2]\) | \(98304\) | \(1.5202\) | |
| 28224.by2 | 28224gd3 | \([0, 0, 0, -104076, 10224144]\) | \(11090466/2401\) | \(26990930365120512\) | \([2]\) | \(196608\) | \(1.8668\) | |
| 28224.by1 | 28224gd4 | \([0, 0, 0, -527436, -147435120]\) | \(1443468546/7\) | \(78690759081984\) | \([2]\) | \(196608\) | \(1.8668\) |
Rank
sage: E.rank()
The elliptic curves in class 28224gd have rank \(0\).
Complex multiplication
The elliptic curves in class 28224gd do not have complex multiplication.Modular form 28224.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
