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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 4730.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4730.c1 | 4730b3 | \([1, -1, 0, -137654, 17939460]\) | \(288464247626448102201/28440150818750000\) | \(28440150818750000\) | \([4]\) | \(30720\) | \(1.8940\) | |
4730.c2 | 4730b2 | \([1, -1, 0, -31174, -1801932]\) | \(3350496292255693881/524098606240000\) | \(524098606240000\) | \([2, 2]\) | \(15360\) | \(1.5474\) | |
4730.c3 | 4730b1 | \([1, -1, 0, -29894, -1981900]\) | \(2954499865542011961/93770547200\) | \(93770547200\) | \([2]\) | \(7680\) | \(1.2009\) | \(\Gamma_0(N)\)-optimal |
4730.c4 | 4730b4 | \([1, -1, 0, 54826, -10040732]\) | \(18225478596775570119/53980968079601200\) | \(-53980968079601200\) | \([2]\) | \(30720\) | \(1.8940\) |
Rank
sage: E.rank()
The elliptic curves in class 4730.c have rank \(1\).
Complex multiplication
The elliptic curves in class 4730.c do not have complex multiplication.Modular form 4730.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.