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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 588.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
588.c1 | 588b4 | \([0, -1, 0, -89588, 10350936]\) | \(2640279346000/3087\) | \(92974710528\) | \([2]\) | \(1728\) | \(1.3878\) | |
588.c2 | 588b3 | \([0, -1, 0, -5553, 165894]\) | \(-10061824000/352947\) | \(-664381785648\) | \([2]\) | \(864\) | \(1.0412\) | |
588.c3 | 588b2 | \([0, -1, 0, -1388, 6840]\) | \(9826000/5103\) | \(153692888832\) | \([2]\) | \(576\) | \(0.83846\) | |
588.c4 | 588b1 | \([0, -1, 0, 327, 666]\) | \(2048000/1323\) | \(-2490394032\) | \([2]\) | \(288\) | \(0.49188\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 588.c have rank \(1\).
Complex multiplication
The elliptic curves in class 588.c do not have complex multiplication.Modular form 588.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.