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SageMath
E = EllipticCurve("y1")
E.isogeny_class()
Elliptic curves in class 221067y
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
221067.z3 | 221067y1 | \([1, -1, 0, -168273, -21544880]\) | \(408023180713/80247321\) | \(103636844669561049\) | \([2]\) | \(1474560\) | \(1.9816\) | \(\Gamma_0(N)\)-optimal |
221067.z2 | 221067y2 | \([1, -1, 0, -827118, 270323455]\) | \(48455467135993/3635004681\) | \(4694492112676562889\) | \([2, 2]\) | \(2949120\) | \(2.3281\) | |
221067.z1 | 221067y3 | \([1, -1, 0, -12985803, 18014708344]\) | \(187519537050946633/1186707753\) | \(1532595051563463657\) | \([2]\) | \(5898240\) | \(2.6747\) | |
221067.z4 | 221067y4 | \([1, -1, 0, 790047, 1197605866]\) | \(42227808999767/504359959257\) | \(-651364732226560539033\) | \([2]\) | \(5898240\) | \(2.6747\) |
Rank
sage: E.rank()
The elliptic curves in class 221067y have rank \(0\).
Complex multiplication
The elliptic curves in class 221067y do not have complex multiplication.Modular form 221067.2.a.y
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.