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SageMath
E = EllipticCurve("qd1")
E.isogeny_class()
Elliptic curves in class 369600.qd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
369600.qd1 | 369600qd4 | \([0, 1, 0, -4959633, 2239858863]\) | \(52702650535889104/22020583921875\) | \(5637269484000000000000\) | \([2]\) | \(23887872\) | \(2.8707\) | |
369600.qd2 | 369600qd2 | \([0, 1, 0, -4275633, 3401470863]\) | \(33766427105425744/9823275\) | \(2514758400000000\) | \([2]\) | \(7962624\) | \(2.3214\) | |
369600.qd3 | 369600qd1 | \([0, 1, 0, -266133, 53538363]\) | \(-130287139815424/2250652635\) | \(-36010442160000000\) | \([2]\) | \(3981312\) | \(1.9748\) | \(\Gamma_0(N)\)-optimal |
369600.qd4 | 369600qd3 | \([0, 1, 0, 1029867, 257334363]\) | \(7549996227362816/6152409907875\) | \(-98438558526000000000\) | \([2]\) | \(11943936\) | \(2.5241\) |
Rank
sage: E.rank()
The elliptic curves in class 369600.qd have rank \(1\).
Complex multiplication
The elliptic curves in class 369600.qd do not have complex multiplication.Modular form 369600.2.a.qd
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.