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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 112710.j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
112710.j1 | 112710j4 | \([1, 1, 0, -25006742, 48121568244]\) | \(71647584155243142409/10140000\) | \(244754949660000\) | \([2]\) | \(6553600\) | \(2.6139\) | |
112710.j2 | 112710j3 | \([1, 1, 0, -1794262, 514103476]\) | \(26465989780414729/10571870144160\) | \(255179245063701947040\) | \([2]\) | \(6553600\) | \(2.6139\) | |
112710.j3 | 112710j2 | \([1, 1, 0, -1563062, 751268436]\) | \(17496824387403529/6580454400\) | \(158836172131353600\) | \([2, 2]\) | \(3276800\) | \(2.2673\) | |
112710.j4 | 112710j1 | \([1, 1, 0, -83382, 15275604]\) | \(-2656166199049/2658140160\) | \(-64161041523671040\) | \([2]\) | \(1638400\) | \(1.9207\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 112710.j have rank \(0\).
Complex multiplication
The elliptic curves in class 112710.j do not have complex multiplication.Modular form 112710.2.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.