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SageMath
E = EllipticCurve("ea1")
E.isogeny_class()
Elliptic curves in class 177870ea
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 177870.fi7 | 177870ea1 | \([1, 1, 1, -6951876, 3204685749]\) | \(178272935636041/81841914000\) | \(17057686506919250346000\) | \([2]\) | \(13271040\) | \(2.9607\) | \(\Gamma_0(N)\)-optimal |
| 177870.fi5 | 177870ea2 | \([1, 1, 1, -93396696, 347185913493]\) | \(432288716775559561/270140062500\) | \(56303234783641215562500\) | \([2, 2]\) | \(26542080\) | \(3.3073\) | |
| 177870.fi4 | 177870ea3 | \([1, 1, 1, -283095051, -1833373145031]\) | \(12038605770121350841/757333463040\) | \(157845242887841513410560\) | \([2]\) | \(39813120\) | \(3.5100\) | |
| 177870.fi2 | 177870ea4 | \([1, 1, 1, -1494122946, 22228771100493]\) | \(1769857772964702379561/691787250\) | \(144183945160224065250\) | \([2]\) | \(53084160\) | \(3.6539\) | |
| 177870.fi6 | 177870ea5 | \([1, 1, 1, -75787566, 482128198509]\) | \(-230979395175477481/348191894531250\) | \(-72570983385901187988281250\) | \([2]\) | \(53084160\) | \(3.6539\) | |
| 177870.fi3 | 177870ea6 | \([1, 1, 1, -300170571, -1599759540807]\) | \(14351050585434661561/3001282273281600\) | \(625534394716275604530062400\) | \([2, 2]\) | \(79626240\) | \(3.8566\) | |
| 177870.fi1 | 177870ea7 | \([1, 1, 1, -1520358771, 21407621083113]\) | \(1864737106103260904761/129177711985836360\) | \(26923526186539356808476236040\) | \([2]\) | \(159252480\) | \(4.2032\) | |
| 177870.fi8 | 177870ea8 | \([1, 1, 1, 646809309, -9655527983991]\) | \(143584693754978072519/276341298967965000\) | \(-57595711247789184664848885000\) | \([2]\) | \(159252480\) | \(4.2032\) |
Rank
sage: E.rank()
The elliptic curves in class 177870ea have rank \(0\).
Complex multiplication
The elliptic curves in class 177870ea do not have complex multiplication.Modular form 177870.2.a.ea
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
