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SageMath
E = EllipticCurve("by1")
E.isogeny_class()
Elliptic curves in class 12870.by
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12870.by1 | 12870bx3 | \([1, -1, 1, -237632, -44527219]\) | \(2035678735521204409/141376950\) | \(103063796550\) | \([2]\) | \(49152\) | \(1.5673\) | |
| 12870.by2 | 12870bx4 | \([1, -1, 1, -25412, 421949]\) | \(2489411558640889/1338278906250\) | \(975605322656250\) | \([2]\) | \(49152\) | \(1.5673\) | |
| 12870.by3 | 12870bx2 | \([1, -1, 1, -14882, -690019]\) | \(499980107400409/4140922500\) | \(3018732502500\) | \([2, 2]\) | \(24576\) | \(1.2207\) | |
| 12870.by4 | 12870bx1 | \([1, -1, 1, -302, -25171]\) | \(-4165509529/375289200\) | \(-273585826800\) | \([2]\) | \(12288\) | \(0.87417\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 12870.by have rank \(0\).
Complex multiplication
The elliptic curves in class 12870.by do not have complex multiplication.Modular form 12870.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
