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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 96330.r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
96330.r1 | 96330u4 | \([1, 1, 0, -104952, -13111626]\) | \(26487576322129/44531250\) | \(214943838281250\) | \([2]\) | \(589824\) | \(1.6454\) | |
96330.r2 | 96330u2 | \([1, 1, 0, -8622, -68544]\) | \(14688124849/8122500\) | \(39205756102500\) | \([2, 2]\) | \(294912\) | \(1.2988\) | |
96330.r3 | 96330u1 | \([1, 1, 0, -5242, 143044]\) | \(3301293169/22800\) | \(110051245200\) | \([2]\) | \(147456\) | \(0.95224\) | \(\Gamma_0(N)\)-optimal |
96330.r4 | 96330u3 | \([1, 1, 0, 33628, -499494]\) | \(871257511151/527800050\) | \(-2547590031540450\) | \([2]\) | \(589824\) | \(1.6454\) |
Rank
sage: E.rank()
The elliptic curves in class 96330.r have rank \(1\).
Complex multiplication
The elliptic curves in class 96330.r do not have complex multiplication.Modular form 96330.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.