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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 166410.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
166410.a1 | 166410by3 | \([1, -1, 0, -13953825, 19929326575]\) | \(65202655558249/512820150\) | \(2363215590957632628150\) | \([2]\) | \(11354112\) | \(2.9294\) | |
166410.a2 | 166410by2 | \([1, -1, 0, -1473075, -172169375]\) | \(76711450249/41602500\) | \(191715705053350402500\) | \([2, 2]\) | \(5677056\) | \(2.5828\) | |
166410.a3 | 166410by1 | \([1, -1, 0, -1140255, -467780099]\) | \(35578826569/51600\) | \(237786920996403600\) | \([2]\) | \(2838528\) | \(2.2362\) | \(\Gamma_0(N)\)-optimal |
166410.a4 | 166410by4 | \([1, -1, 0, 5682555, -1358572829]\) | \(4403686064471/2721093750\) | \(-12539544661919721093750\) | \([2]\) | \(11354112\) | \(2.9294\) |
Rank
sage: E.rank()
The elliptic curves in class 166410.a have rank \(1\).
Complex multiplication
The elliptic curves in class 166410.a do not have complex multiplication.Modular form 166410.2.a.a
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.