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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 473382x
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
473382.x2 | 473382x1 | \([1, -1, 0, 1857060, 11005257424]\) | \(40251338884511/2997011332224\) | \(-52736273944671929399424\) | \([]\) | \(47416320\) | \(3.0395\) | \(\Gamma_0(N)\)-optimal* |
473382.x1 | 473382x2 | \([1, -1, 0, -9557364150, 359631893307694]\) | \(-5486773802537974663600129/2635437714\) | \(-46373920497153526914\) | \([]\) | \(331914240\) | \(4.0125\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 473382x have rank \(0\).
Complex multiplication
The elliptic curves in class 473382x do not have complex multiplication.Modular form 473382.2.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.
\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.