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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 972.b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
|---|---|---|---|---|---|---|---|---|---|
| 972.b1 | 972c2 | \([0, 0, 0, 0, -243]\) | \(0\) | \(-25509168\) | \([]\) | \(162\) | \(0.10009\) | \(-3\) | |
| 972.b2 | 972c1 | \([0, 0, 0, 0, 9]\) | \(0\) | \(-34992\) | \([3]\) | \(54\) | \(-0.44921\) | \(\Gamma_0(N)\)-optimal | \(-3\) |
Rank
sage: E.rank()
The elliptic curves in class 972.b have rank \(1\).
Complex multiplication
Each elliptic curve in class 972.b has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).Modular form 972.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
