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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 42432j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
42432.s2 | 42432j1 | \([0, -1, 0, -160433, -24264015]\) | \(27873248949250000/538367795433\) | \(8820617960374272\) | \([2]\) | \(294912\) | \(1.8525\) | \(\Gamma_0(N)\)-optimal |
42432.s1 | 42432j2 | \([0, -1, 0, -336193, 38622913]\) | \(64122592551794500/27331783704693\) | \(1791215776870760448\) | \([2]\) | \(589824\) | \(2.1990\) |
Rank
sage: E.rank()
The elliptic curves in class 42432j have rank \(1\).
Complex multiplication
The elliptic curves in class 42432j do not have complex multiplication.Modular form 42432.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.