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SageMath
E = EllipticCurve("by1")
E.isogeny_class()
Elliptic curves in class 20286by
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 20286.bz4 | 20286by1 | \([1, -1, 1, 2048215, 7945492713]\) | \(11079872671250375/324440155855872\) | \(-27825973664393576512512\) | \([2]\) | \(1843200\) | \(2.9867\) | \(\Gamma_0(N)\)-optimal |
| 20286.bz2 | 20286by2 | \([1, -1, 1, -49390025, 127302784809]\) | \(155355156733986861625/8291568305839392\) | \(711135650598386300838432\) | \([2]\) | \(3686400\) | \(3.3333\) | |
| 20286.bz3 | 20286by3 | \([1, -1, 1, -18491360, -218306963325]\) | \(-8152944444844179625/235342826399858688\) | \(-20184441325492274617909248\) | \([2]\) | \(5529600\) | \(3.5360\) | |
| 20286.bz1 | 20286by4 | \([1, -1, 1, -668772320, -6624354756477]\) | \(385693937170561837203625/2159357734550274048\) | \(185199736743724684583927808\) | \([2]\) | \(11059200\) | \(3.8826\) |
Rank
sage: E.rank()
The elliptic curves in class 20286by have rank \(0\).
Complex multiplication
The elliptic curves in class 20286by do not have complex multiplication.Modular form 20286.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
