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SageMath
E = EllipticCurve("it1")
E.isogeny_class()
Elliptic curves in class 221760.it
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 221760.it1 | 221760br7 | \([0, 0, 0, -147702252, 648250424656]\) | \(1864737106103260904761/129177711985836360\) | \(24686245193364198245007360\) | \([4]\) | \(42467328\) | \(3.6203\) | |
| 221760.it2 | 221760br4 | \([0, 0, 0, -145153452, 673114841296]\) | \(1769857772964702379561/691787250\) | \(132202602233856000\) | \([4]\) | \(14155776\) | \(3.0710\) | |
| 221760.it3 | 221760br6 | \([0, 0, 0, -29161452, -48437565104]\) | \(14351050585434661561/3001282273281600\) | \(573553974240159046041600\) | \([2, 2]\) | \(21233664\) | \(3.2737\) | |
| 221760.it4 | 221760br3 | \([0, 0, 0, -27502572, -55511692976]\) | \(12038605770121350841/757333463040\) | \(144728678611330007040\) | \([2]\) | \(10616832\) | \(2.9272\) | |
| 221760.it5 | 221760br2 | \([0, 0, 0, -9073452, 10514105296]\) | \(432288716775559561/270140062500\) | \(51624569880576000000\) | \([2, 2]\) | \(7077888\) | \(2.7244\) | |
| 221760.it6 | 221760br5 | \([0, 0, 0, -7362732, 14599988944]\) | \(-230979395175477481/348191894531250\) | \(-66540507264000000000000\) | \([2]\) | \(14155776\) | \(3.0710\) | |
| 221760.it7 | 221760br1 | \([0, 0, 0, -675372, 97126864]\) | \(178272935636041/81841914000\) | \(15640233326936064000\) | \([2]\) | \(3538944\) | \(2.3779\) | \(\Gamma_0(N)\)-optimal |
| 221760.it8 | 221760br8 | \([0, 0, 0, 62837268, -292381371056]\) | \(143584693754978072519/276341298967965000\) | \(-52809644624483840163840000\) | \([2]\) | \(42467328\) | \(3.6203\) |
Rank
sage: E.rank()
The elliptic curves in class 221760.it have rank \(1\).
Complex multiplication
The elliptic curves in class 221760.it do not have complex multiplication.Modular form 221760.2.a.it
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
