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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 16758.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
16758.i1 | 16758g3 | \([1, -1, 0, -188757, -31517375]\) | \(8671983378625/82308\) | \(7059237887268\) | \([2]\) | \(82944\) | \(1.6278\) | |
16758.i2 | 16758g4 | \([1, -1, 0, -184347, -33063521]\) | \(-8078253774625/846825858\) | \(-72628969003156818\) | \([2]\) | \(165888\) | \(1.9743\) | |
16758.i3 | 16758g1 | \([1, -1, 0, -3537, 7069]\) | \(57066625/32832\) | \(2815873284672\) | \([2]\) | \(27648\) | \(1.0785\) | \(\Gamma_0(N)\)-optimal |
16758.i4 | 16758g2 | \([1, -1, 0, 14103, 45877]\) | \(3616805375/2105352\) | \(-180567874379592\) | \([2]\) | \(55296\) | \(1.4250\) |
Rank
sage: E.rank()
The elliptic curves in class 16758.i have rank \(1\).
Complex multiplication
The elliptic curves in class 16758.i do not have complex multiplication.Modular form 16758.2.a.i
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.