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SageMath
sage: E = EllipticCurve("bd1")
sage: E.isogeny_class()
Elliptic curves in class 28224.bd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 28224.bd1 | 28224ck4 | \([0, 0, 0, -527436, 147435120]\) | \(1443468546/7\) | \(78690759081984\) | \([2]\) | \(196608\) | \(1.8668\) | |
| 28224.bd2 | 28224ck3 | \([0, 0, 0, -104076, -10224144]\) | \(11090466/2401\) | \(26990930365120512\) | \([2]\) | \(196608\) | \(1.8668\) | |
| 28224.bd3 | 28224ck2 | \([0, 0, 0, -33516, 2222640]\) | \(740772/49\) | \(275417656786944\) | \([2, 2]\) | \(98304\) | \(1.5202\) | |
| 28224.bd4 | 28224ck1 | \([0, 0, 0, 1764, 148176]\) | \(432/7\) | \(-9836344885248\) | \([2]\) | \(49152\) | \(1.1737\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 28224.bd have rank \(1\).
Complex multiplication
The elliptic curves in class 28224.bd do not have complex multiplication.Modular form 28224.2.a.bd
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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
