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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 4830.z
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4830.z1 | 4830y4 | \([1, 1, 1, -36920100, 86327581317]\) | \(5565604209893236690185614401/229307220930246900000\) | \(229307220930246900000\) | \([2]\) | \(614400\) | \(2.9882\) | |
4830.z2 | 4830y3 | \([1, 1, 1, -11255780, -13408567675]\) | \(157706830105239346386477121/13650704956054687500000\) | \(13650704956054687500000\) | \([2]\) | \(614400\) | \(2.9882\) | |
4830.z3 | 4830y2 | \([1, 1, 1, -2420100, 1209181317]\) | \(1567558142704512417614401/274462175610000000000\) | \(274462175610000000000\) | \([2, 2]\) | \(307200\) | \(2.6416\) | |
4830.z4 | 4830y1 | \([1, 1, 1, 288380, 108455045]\) | \(2652277923951208297919/6605028468326400000\) | \(-6605028468326400000\) | \([4]\) | \(153600\) | \(2.2950\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 4830.z have rank \(1\).
Complex multiplication
The elliptic curves in class 4830.z do not have complex multiplication.Modular form 4830.2.a.z
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.