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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 208080.j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
208080.j1 | 208080ft3 | \([0, 0, 0, -70155150843, 7152162991285642]\) | \(1059623036730633329075378/154307373046875\) | \(5560805265303721500000000000\) | \([2]\) | \(495452160\) | \(4.7323\) | |
208080.j2 | 208080ft4 | \([0, 0, 0, -8155842123, -106919462654822]\) | \(1664865424893526702418/826424127435466125\) | \(29782009430107672555696621824000\) | \([2]\) | \(495452160\) | \(4.7323\) | |
208080.j3 | 208080ft2 | \([0, 0, 0, -4397397123, 111072602412178]\) | \(521902963282042184836/6241849278890625\) | \(112469377353547521536784000000\) | \([2, 2]\) | \(247726080\) | \(4.3857\) | |
208080.j4 | 208080ft1 | \([0, 0, 0, -52634703, 4461691102702]\) | \(-3579968623693264/1906997690433375\) | \(-8590356530357240554434144000\) | \([2]\) | \(123863040\) | \(4.0391\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 208080.j have rank \(0\).
Complex multiplication
The elliptic curves in class 208080.j do not have complex multiplication.Modular form 208080.2.a.j
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.