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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 22050r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
22050.cv2 | 22050r1 | \([1, -1, 0, -134367, -8105959]\) | \(59319/28\) | \(126639038039062500\) | \([2]\) | \(276480\) | \(1.9758\) | \(\Gamma_0(N)\)-optimal |
22050.cv1 | 22050r2 | \([1, -1, 0, -1788117, -919322209]\) | \(139798359/98\) | \(443236633136718750\) | \([2]\) | \(552960\) | \(2.3224\) |
Rank
sage: E.rank()
The elliptic curves in class 22050r have rank \(1\).
Complex multiplication
The elliptic curves in class 22050r do not have complex multiplication.Modular form 22050.2.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 Cremona numbering.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.