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SageMath
E = EllipticCurve("y1")
E.isogeny_class()
Elliptic curves in class 473200y
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
473200.y4 | 473200y1 | \([0, 1, 0, -15786008, -24114428012]\) | \(1408317602329/2153060\) | \(665114200674560000000\) | \([2]\) | \(27869184\) | \(2.8952\) | \(\Gamma_0(N)\)-optimal* |
473200.y3 | 473200y2 | \([0, 1, 0, -20518008, -8470436012]\) | \(3092354182009/1689383150\) | \(521877106743574400000000\) | \([2]\) | \(55738368\) | \(3.2418\) | \(\Gamma_0(N)\)-optimal* |
473200.y2 | 473200y3 | \([0, 1, 0, -64120008, 174063759988]\) | \(94376601570889/12235496000\) | \(3779737741584896000000000\) | \([2]\) | \(83607552\) | \(3.4445\) | \(\Gamma_0(N)\)-optimal* |
473200.y1 | 473200y4 | \([0, 1, 0, -991592008, 12017881199988]\) | \(349046010201856969/7245875000\) | \(2238365098424000000000000\) | \([2]\) | \(167215104\) | \(3.7911\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 473200y have rank \(0\).
Complex multiplication
The elliptic curves in class 473200y do not have complex multiplication.Modular form 473200.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.