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SageMath
E = EllipticCurve("js1")
E.isogeny_class()
Elliptic curves in class 221760.js
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
221760.js1 | 221760jl5 | \([0, 0, 0, -7632012, 8114950384]\) | \(257260669489908001/14267882475\) | \(2726634802190745600\) | \([2]\) | \(8388608\) | \(2.6029\) | |
221760.js2 | 221760jl3 | \([0, 0, 0, -504012, 111631984]\) | \(74093292126001/14707625625\) | \(2810671026831360000\) | \([2, 2]\) | \(4194304\) | \(2.2563\) | |
221760.js3 | 221760jl2 | \([0, 0, 0, -155532, -22044944]\) | \(2177286259681/161417025\) | \(30847273854566400\) | \([2, 2]\) | \(2097152\) | \(1.9097\) | |
221760.js4 | 221760jl1 | \([0, 0, 0, -152652, -22956176]\) | \(2058561081361/12705\) | \(2427963310080\) | \([2]\) | \(1048576\) | \(1.5632\) | \(\Gamma_0(N)\)-optimal |
221760.js5 | 221760jl4 | \([0, 0, 0, 146868, -97403024]\) | \(1833318007919/22507682505\) | \(-4301285109568634880\) | \([2]\) | \(4194304\) | \(2.2563\) | |
221760.js6 | 221760jl6 | \([0, 0, 0, 1048308, 663636976]\) | \(666688497209279/1381398046875\) | \(-263989277798400000000\) | \([2]\) | \(8388608\) | \(2.6029\) |
Rank
sage: E.rank()
The elliptic curves in class 221760.js have rank \(0\).
Complex multiplication
The elliptic curves in class 221760.js do not have complex multiplication.Modular form 221760.2.a.js
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.