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SageMath
E = EllipticCurve("by1")
E.isogeny_class()
Elliptic curves in class 123690by
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 123690.by4 | 123690by1 | \([1, 0, 0, 583290, 1302659172]\) | \(21947105851758112102559/745716001590000000000\) | \(-745716001590000000000\) | \([10]\) | \(7680000\) | \(2.6850\) | \(\Gamma_0(N)\)-optimal |
| 123690.by3 | 123690by2 | \([1, 0, 0, -14916710, 21176759172]\) | \(367064896520710054039897441/18666454802704739100000\) | \(18666454802704739100000\) | \([10]\) | \(15360000\) | \(3.0316\) | |
| 123690.by2 | 123690by3 | \([1, 0, 0, -94347210, -366455113728]\) | \(-92877581523634426145841409441/4261909641706236188127900\) | \(-4261909641706236188127900\) | \([2]\) | \(38400000\) | \(3.4897\) | |
| 123690.by1 | 123690by4 | \([1, 0, 0, -1525804760, -22940254385718]\) | \(392844437751976119681038861408641/774075047633450828153910\) | \(774075047633450828153910\) | \([2]\) | \(76800000\) | \(3.8363\) |
Rank
sage: E.rank()
The elliptic curves in class 123690by have rank \(0\).
Complex multiplication
The elliptic curves in class 123690by do not have complex multiplication.Modular form 123690.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
