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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 468270d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
468270.d4 | 468270d1 | \([1, -1, 0, -117090, 4600800]\) | \(137467988281/72562500\) | \(93712144500562500\) | \([2]\) | \(4147200\) | \(1.9481\) | \(\Gamma_0(N)\)-optimal* |
468270.d3 | 468270d2 | \([1, -1, 0, -1478340, 691487550]\) | \(276670733768281/336980250\) | \(435199199060612250\) | \([2]\) | \(8294400\) | \(2.2947\) | \(\Gamma_0(N)\)-optimal* |
468270.d2 | 468270d3 | \([1, -1, 0, -5425965, -4863310875]\) | \(13679527032530281/381633600\) | \(492867570294158400\) | \([2]\) | \(12441600\) | \(2.4975\) | |
468270.d1 | 468270d4 | \([1, -1, 0, -5643765, -4451538195]\) | \(15393836938735081/2275690697640\) | \(2938981643353323893160\) | \([2]\) | \(24883200\) | \(2.8440\) |
Rank
sage: E.rank()
The elliptic curves in class 468270d have rank \(1\).
Complex multiplication
The elliptic curves in class 468270d do not have complex multiplication.Modular form 468270.2.a.d
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.