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SageMath
E = EllipticCurve("ce1")
E.isogeny_class()
Elliptic curves in class 78400.ce
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
78400.ce1 | 78400iq2 | \([0, 1, 0, -38496033, -91949611937]\) | \(-5452947409/250\) | \(-289254654976000000000\) | \([]\) | \(5806080\) | \(3.0018\) | |
78400.ce2 | 78400iq1 | \([0, 1, 0, -80033, -327451937]\) | \(-49/40\) | \(-46280744796160000000\) | \([]\) | \(1935360\) | \(2.4525\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 78400.ce have rank \(1\).
Complex multiplication
The elliptic curves in class 78400.ce do not have complex multiplication.Modular form 78400.2.a.ce
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.