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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 133952x
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 133952.br3 | 133952x1 | \([0, 1, 0, 64843, -6787711]\) | \(471114356703100928/585612268875179\) | \(-37479185208011456\) | \([]\) | \(1119744\) | \(1.8663\) | \(\Gamma_0(N)\)-optimal |
| 133952.br2 | 133952x2 | \([0, 1, 0, -638197, 305274169]\) | \(-449167881463536812032/369990050199923699\) | \(-23679363212795116736\) | \([]\) | \(3359232\) | \(2.4156\) | |
| 133952.br1 | 133952x3 | \([0, 1, 0, -59318637, 175827209849]\) | \(-360675992659311050823073792/56219378022244619\) | \(-3598040193423655616\) | \([]\) | \(10077696\) | \(2.9649\) |
Rank
sage: E.rank()
The elliptic curves in class 133952x have rank \(1\).
Complex multiplication
The elliptic curves in class 133952x do not have complex multiplication.Modular form 133952.2.a.x
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.
