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SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 342225.bp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
342225.bp1 | 342225bp2 | \([0, 0, 1, -684450, -216885094]\) | \(884736/5\) | \(200403453263203125\) | \([]\) | \(3732480\) | \(2.1619\) | |
342225.bp2 | 342225bp1 | \([0, 0, 1, -50700, 4188031]\) | \(2359296/125\) | \(763616267578125\) | \([]\) | \(1244160\) | \(1.6126\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 342225.bp have rank \(1\).
Complex multiplication
The elliptic curves in class 342225.bp do not have complex multiplication.Modular form 342225.2.a.bp
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.