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SageMath
E = EllipticCurve("bt1")
E.isogeny_class()
Elliptic curves in class 342225bt
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
342225.bt2 | 342225bt1 | \([0, 0, 1, -76050, 8032781]\) | \(884736/5\) | \(274901856328125\) | \([]\) | \(1244160\) | \(1.6126\) | \(\Gamma_0(N)\)-optimal |
342225.bt1 | 342225bt2 | \([0, 0, 1, -456300, -113076844]\) | \(2359296/125\) | \(556676259064453125\) | \([]\) | \(3732480\) | \(2.1619\) |
Rank
sage: E.rank()
The elliptic curves in class 342225bt have rank \(1\).
Complex multiplication
The elliptic curves in class 342225bt do not have complex multiplication.Modular form 342225.2.a.bt
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.