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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 215296b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
215296.c2 | 215296b1 | \([0, 1, 0, -2803, -51519]\) | \(8000\) | \(304549540352\) | \([2]\) | \(201600\) | \(0.93310\) | \(\Gamma_0(N)\)-optimal | \(-8\) |
215296.c1 | 215296b2 | \([0, 1, 0, -11213, 400939]\) | \(8000\) | \(19491170582528\) | \([2]\) | \(403200\) | \(1.2797\) | \(-8\) |
Rank
sage: E.rank()
The elliptic curves in class 215296b have rank \(0\).
Complex multiplication
Each elliptic curve in class 215296b has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-2}) \).Modular form 215296.2.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 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.