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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 127050bd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 127050.by7 | 127050bd1 | \([1, 1, 0, -3546875, -1170421875]\) | \(178272935636041/81841914000\) | \(2265436609496156250000\) | \([2]\) | \(6635520\) | \(2.7925\) | \(\Gamma_0(N)\)-optimal |
| 127050.by5 | 127050bd2 | \([1, 1, 0, -47651375, -126559515375]\) | \(432288716775559561/270140062500\) | \(7477649988477539062500\) | \([2, 2]\) | \(13271040\) | \(3.1391\) | |
| 127050.by4 | 127050bd3 | \([1, 1, 0, -144436250, 668035732500]\) | \(12038605770121350841/757333463040\) | \(20963475423696960000000\) | \([2]\) | \(19906560\) | \(3.3418\) | |
| 127050.by6 | 127050bd4 | \([1, 1, 0, -38667125, -175730315625]\) | \(-230979395175477481/348191894531250\) | \(-9638174701057434082031250\) | \([2]\) | \(26542080\) | \(3.4856\) | |
| 127050.by2 | 127050bd5 | \([1, 1, 0, -762307625, -8101408609125]\) | \(1769857772964702379561/691787250\) | \(19149114256207031250\) | \([2]\) | \(26542080\) | \(3.4856\) | |
| 127050.by3 | 127050bd6 | \([1, 1, 0, -153148250, 582893356500]\) | \(14351050585434661561/3001282273281600\) | \(83077416020891009025000000\) | \([2, 2]\) | \(39813120\) | \(3.6884\) | |
| 127050.by8 | 127050bd7 | \([1, 1, 0, 330004750, 3519014137500]\) | \(143584693754978072519/276341298967965000\) | \(-7649304186577922552578125000\) | \([2]\) | \(79626240\) | \(4.0350\) | |
| 127050.by1 | 127050bd8 | \([1, 1, 0, -775693250, -7802165248500]\) | \(1864737106103260904761/129177711985836360\) | \(3575721822239691371218125000\) | \([2]\) | \(79626240\) | \(4.0350\) |
Rank
sage: E.rank()
The elliptic curves in class 127050bd have rank \(1\).
Complex multiplication
The elliptic curves in class 127050bd do not have complex multiplication.Modular form 127050.2.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 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.
