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SageMath
E = EllipticCurve("ew1")
E.isogeny_class()
Elliptic curves in class 388080ew
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
388080.ew3 | 388080ew1 | \([0, 0, 0, -50568, -353633]\) | \(281370820608/161767375\) | \(8221724597394000\) | \([2]\) | \(1990656\) | \(1.7434\) | \(\Gamma_0(N)\)-optimal |
388080.ew4 | 388080ew2 | \([0, 0, 0, 201537, -2824262]\) | \(1113258734352/648484375\) | \(-527340936276000000\) | \([2]\) | \(3981312\) | \(2.0900\) | |
388080.ew1 | 388080ew3 | \([0, 0, 0, -2931768, -1932150213]\) | \(75216478666752/326095\) | \(12082134194277840\) | \([2]\) | \(5971968\) | \(2.2927\) | |
388080.ew2 | 388080ew4 | \([0, 0, 0, -2885463, -1996134462]\) | \(-4481782160112/310023175\) | \(-183786521286672057600\) | \([2]\) | \(11943936\) | \(2.6393\) |
Rank
sage: E.rank()
The elliptic curves in class 388080ew have rank \(0\).
Complex multiplication
The elliptic curves in class 388080ew do not have complex multiplication.Modular form 388080.2.a.ew
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 Cremona 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 Cremona labels.