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SageMath
E = EllipticCurve("bz1")
E.isogeny_class()
Elliptic curves in class 112896.bz
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
112896.bz1 | 112896dm2 | \([0, 0, 0, -5880, 153664]\) | \(8000\) | \(2810384252928\) | \([2]\) | \(147456\) | \(1.1183\) | \(-8\) | |
112896.bz2 | 112896dm1 | \([0, 0, 0, -1470, -19208]\) | \(8000\) | \(43912253952\) | \([2]\) | \(73728\) | \(0.77172\) | \(\Gamma_0(N)\)-optimal | \(-8\) |
Rank
sage: E.rank()
The elliptic curves in class 112896.bz have rank \(2\).
Complex multiplication
Each elliptic curve in class 112896.bz has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-2}) \).Modular form 112896.2.a.bz
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.