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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 47190.r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
47190.r1 | 47190r4 | \([1, 1, 0, -784687, 267194371]\) | \(30161840495801041/2799263610\) | \(4959066240195210\) | \([2]\) | \(983040\) | \(2.0498\) | |
47190.r2 | 47190r3 | \([1, 1, 0, -288587, -56846049]\) | \(1500376464746641/83599963590\) | \(148102435097463990\) | \([2]\) | \(983040\) | \(2.0498\) | |
47190.r3 | 47190r2 | \([1, 1, 0, -52637, 3509961]\) | \(9104453457841/2226896100\) | \(3945082281812100\) | \([2, 2]\) | \(491520\) | \(1.7033\) | |
47190.r4 | 47190r1 | \([1, 1, 0, 7863, 351861]\) | \(30342134159/47190000\) | \(-83599963590000\) | \([2]\) | \(245760\) | \(1.3567\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 47190.r have rank \(1\).
Complex multiplication
The elliptic curves in class 47190.r do not have complex multiplication.Modular form 47190.2.a.r
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.