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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 162450.i
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 162450.i1 | 162450dg4 | \([1, -1, 0, -37625939667, 1793801850972741]\) | \(10993009831928446009969/3767761230468750000\) | \(2019075379493731498718261718750000\) | \([2]\) | \(1194393600\) | \(5.0926\) | |
| 162450.i2 | 162450dg2 | \([1, -1, 0, -33707645667, 2382002636910741]\) | \(7903870428425797297009/886464000000\) | \(475040090845899000000000000\) | \([2]\) | \(398131200\) | \(4.5433\) | |
| 162450.i3 | 162450dg1 | \([1, -1, 0, -2101373667, 37417773678741]\) | \(-1914980734749238129/20440940544000\) | \(-10953931860738146304000000000\) | \([2]\) | \(199065600\) | \(4.1967\) | \(\Gamma_0(N)\)-optimal |
| 162450.i4 | 162450dg3 | \([1, -1, 0, 6943842333, 194771782158741]\) | \(69096190760262356111/70568821500000000\) | \(-37816560374981985210937500000000\) | \([2]\) | \(597196800\) | \(4.7460\) |
Rank
sage: E.rank()
The elliptic curves in class 162450.i have rank \(1\).
Complex multiplication
The elliptic curves in class 162450.i do not have complex multiplication.Modular form 162450.2.a.i
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
