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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 43681j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
43681.i3 | 43681j1 | \([0, -1, 1, 29121, -94238]\) | \(32768/19\) | \(-1583548311814579\) | \([]\) | \(162000\) | \(1.6060\) | \(\Gamma_0(N)\)-optimal |
43681.i2 | 43681j2 | \([0, -1, 1, -407689, -106457473]\) | \(-89915392/6859\) | \(-571660940565063019\) | \([]\) | \(486000\) | \(2.1553\) | |
43681.i1 | 43681j3 | \([0, -1, 1, -33605249, -74971104968]\) | \(-50357871050752/19\) | \(-1583548311814579\) | \([]\) | \(1458000\) | \(2.7046\) |
Rank
sage: E.rank()
The elliptic curves in class 43681j have rank \(1\).
Complex multiplication
The elliptic curves in class 43681j do not have complex multiplication.Modular form 43681.2.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 Cremona numbering.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.