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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 250470bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
250470.bc2 | 250470bc1 | \([1, -1, 0, 4515, 2186245]\) | \(212776173/43335680\) | \(-2072838616104960\) | \([2]\) | \(1254400\) | \(1.6182\) | \(\Gamma_0(N)\)-optimal |
250470.bc1 | 250470bc2 | \([1, -1, 0, -227805, 40704901]\) | \(27333463470867/895491200\) | \(42833266715606400\) | \([2]\) | \(2508800\) | \(1.9648\) |
Rank
sage: E.rank()
The elliptic curves in class 250470bc have rank \(1\).
Complex multiplication
The elliptic curves in class 250470bc do not have complex multiplication.Modular form 250470.2.a.bc
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.