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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 331200.w
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
331200.w1 | 331200w3 | \([0, 0, 0, -5918700, 5529166000]\) | \(7679186557489/20988075\) | \(62670056140800000000\) | \([2]\) | \(15728640\) | \(2.6721\) | |
331200.w2 | 331200w4 | \([0, 0, 0, -5486700, -4927826000]\) | \(6117442271569/26953125\) | \(80481600000000000000\) | \([2]\) | \(15728640\) | \(2.6721\) | |
331200.w3 | 331200w2 | \([0, 0, 0, -518700, 10366000]\) | \(5168743489/2975625\) | \(8885168640000000000\) | \([2, 2]\) | \(7864320\) | \(2.3255\) | |
331200.w4 | 331200w1 | \([0, 0, 0, 129300, 1294000]\) | \(80062991/46575\) | \(-139072204800000000\) | \([2]\) | \(3932160\) | \(1.9789\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 331200.w have rank \(0\).
Complex multiplication
The elliptic curves in class 331200.w do not have complex multiplication.Modular form 331200.2.a.w
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.