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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 1690.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1690.b1 | 1690a3 | \([1, -1, 0, -234350, -43607680]\) | \(294889639316481/260\) | \(1254970340\) | \([2]\) | \(5376\) | \(1.4783\) | |
1690.b2 | 1690a2 | \([1, -1, 0, -14650, -678300]\) | \(72043225281/67600\) | \(326292288400\) | \([2, 2]\) | \(2688\) | \(1.1317\) | |
1690.b3 | 1690a4 | \([1, -1, 0, -11270, -1002104]\) | \(-32798729601/71402500\) | \(-344646229622500\) | \([2]\) | \(5376\) | \(1.4783\) | |
1690.b4 | 1690a1 | \([1, -1, 0, -1130, -5004]\) | \(33076161/16640\) | \(80318101760\) | \([2]\) | \(1344\) | \(0.78511\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1690.b have rank \(1\).
Complex multiplication
The elliptic curves in class 1690.b do not have complex multiplication.Modular form 1690.2.a.b
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.