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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 630.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
630.c1 | 630a3 | \([1, -1, 0, -420, -2800]\) | \(416832723/56000\) | \(1102248000\) | \([2]\) | \(288\) | \(0.46251\) | |
630.c2 | 630a1 | \([1, -1, 0, -105, 441]\) | \(4767078987/6860\) | \(185220\) | \([6]\) | \(96\) | \(-0.086795\) | \(\Gamma_0(N)\)-optimal |
630.c3 | 630a2 | \([1, -1, 0, -75, 675]\) | \(-1740992427/5882450\) | \(-158826150\) | \([6]\) | \(192\) | \(0.25978\) | |
630.c4 | 630a4 | \([1, -1, 0, 660, -15544]\) | \(1613964717/6125000\) | \(-120558375000\) | \([2]\) | \(576\) | \(0.80908\) |
Rank
sage: E.rank()
The elliptic curves in class 630.c have rank \(0\).
Complex multiplication
The elliptic curves in class 630.c do not have complex multiplication.Modular form 630.2.a.c
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.