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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 1849.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
1849.b1 | 1849a2 | \([0, 0, 1, -1590140, -771794326]\) | \(-884736000\) | \(-502592611936843\) | \([]\) | \(11352\) | \(2.1060\) | \(-43\) | |
1849.b2 | 1849a1 | \([0, 0, 1, -860, 9707]\) | \(-884736000\) | \(-79507\) | \([]\) | \(264\) | \(0.22543\) | \(\Gamma_0(N)\)-optimal | \(-43\) |
Rank
sage: E.rank()
The elliptic curves in class 1849.b have rank \(1\).
Complex multiplication
Each elliptic curve in class 1849.b has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-43}) \).Modular form 1849.2.a.b
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.
\(\left(\begin{array}{rr} 1 & 43 \\ 43 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.