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SageMath
E = EllipticCurve("ey1")
E.isogeny_class()
Elliptic curves in class 428400.ey
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
428400.ey1 | 428400ey2 | \([0, 0, 0, -135660675, -607419130750]\) | \(5918043195362419129/8515734343200\) | \(397310101516339200000000\) | \([2]\) | \(70778880\) | \(3.4307\) | \(\Gamma_0(N)\)-optimal* |
428400.ey2 | 428400ey1 | \([0, 0, 0, -6060675, -15017530750]\) | \(-527690404915129/1782829440000\) | \(-83179690352640000000000\) | \([2]\) | \(35389440\) | \(3.0841\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 428400.ey have rank \(1\).
Complex multiplication
The elliptic curves in class 428400.ey do not have complex multiplication.Modular form 428400.2.a.ey
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.