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SageMath
E = EllipticCurve("by1")
E.isogeny_class()
Elliptic curves in class 371280by
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 371280.by7 | 371280by1 | \([0, -1, 0, -1015560, 282063600]\) | \(28280100765151839241/7994847656250000\) | \(32746896000000000000\) | \([2]\) | \(7299072\) | \(2.4515\) | \(\Gamma_0(N)\)-optimal |
| 371280.by6 | 371280by2 | \([0, -1, 0, -6015560, -5453936400]\) | \(5877491705974396839241/261806444735062500\) | \(1072359197634816000000\) | \([2, 2]\) | \(14598144\) | \(2.7980\) | |
| 371280.by4 | 371280by3 | \([0, -1, 0, -75565560, 252858303600]\) | \(11650256451486052494789241/580277967360000\) | \(2376818554306560000\) | \([2]\) | \(21897216\) | \(3.0008\) | |
| 371280.by8 | 371280by4 | \([0, -1, 0, 3164440, -20626640400]\) | \(855567391070976980759/45363085180055574750\) | \(-185807196897507634176000\) | \([2]\) | \(29196288\) | \(3.1446\) | |
| 371280.by2 | 371280by5 | \([0, -1, 0, -95195560, -357465232400]\) | \(23292378980986805290659241/49479832772574750\) | \(202669395036466176000\) | \([2]\) | \(29196288\) | \(3.1446\) | |
| 371280.by3 | 371280by6 | \([0, -1, 0, -75693560, 251958822000]\) | \(11709559667189768059461241/82207646338733697600\) | \(336722519403453225369600\) | \([2, 2]\) | \(43794432\) | \(3.3473\) | |
| 371280.by5 | 371280by7 | \([0, -1, 0, -28528760, 562265474160]\) | \(-626920492174472718626041/32979221374608565962360\) | \(-135082890750396686181826560\) | \([2]\) | \(87588864\) | \(3.6939\) | |
| 371280.by1 | 371280by8 | \([0, -1, 0, -124906360, -115916700560]\) | \(52615951054626272117608441/29030877531795041917560\) | \(118910474370232491694325760\) | \([2]\) | \(87588864\) | \(3.6939\) |
Rank
sage: E.rank()
The elliptic curves in class 371280by have rank \(0\).
Complex multiplication
The elliptic curves in class 371280by do not have complex multiplication.Modular form 371280.2.a.by
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.
