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SageMath
E = EllipticCurve("dy1")
E.isogeny_class()
Elliptic curves in class 333270.dy
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
333270.dy1 | 333270dy3 | \([1, -1, 1, -8777003, 9346920581]\) | \(692895692874169/51420783750\) | \(5549236526485334733750\) | \([2]\) | \(25952256\) | \(2.9181\) | |
333270.dy2 | 333270dy2 | \([1, -1, 1, -1778333, -739562623]\) | \(5763259856089/1143116100\) | \(123362989700316704100\) | \([2, 2]\) | \(12976128\) | \(2.5715\) | |
333270.dy3 | 333270dy1 | \([1, -1, 1, -1683113, -840000679]\) | \(4886171981209/270480\) | \(29189704750148880\) | \([2]\) | \(6488064\) | \(2.2249\) | \(\Gamma_0(N)\)-optimal |
333270.dy4 | 333270dy4 | \([1, -1, 1, 3696817, -4399152883]\) | \(51774168853511/107398242630\) | \(-11590221062757146343030\) | \([2]\) | \(25952256\) | \(2.9181\) |
Rank
sage: E.rank()
The elliptic curves in class 333270.dy have rank \(1\).
Complex multiplication
The elliptic curves in class 333270.dy do not have complex multiplication.Modular form 333270.2.a.dy
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.