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SageMath
E = EllipticCurve("eu1")
E.isogeny_class()
Elliptic curves in class 476850eu
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
476850.eu1 | 476850eu1 | \([1, 0, 1, -1430701, 658530548]\) | \(858729462625/38148\) | \(14387499722062500\) | \([2]\) | \(10616832\) | \(2.1777\) | \(\Gamma_0(N)\)-optimal |
476850.eu2 | 476850eu2 | \([1, 0, 1, -1358451, 728035048]\) | \(-735091890625/181908738\) | \(-68606792424655031250\) | \([2]\) | \(21233664\) | \(2.5243\) |
Rank
sage: E.rank()
The elliptic curves in class 476850eu have rank \(2\).
Complex multiplication
The elliptic curves in class 476850eu do not have complex multiplication.Modular form 476850.2.a.eu
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.