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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 900.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
900.d1 | 900c2 | \([0, 0, 0, 0, -2700]\) | \(0\) | \(-3149280000\) | \([]\) | \(432\) | \(0.50142\) | \(-3\) | |
900.d2 | 900c1 | \([0, 0, 0, 0, 100]\) | \(0\) | \(-4320000\) | \([3]\) | \(144\) | \(-0.047887\) | \(\Gamma_0(N)\)-optimal | \(-3\) |
Rank
sage: E.rank()
The elliptic curves in class 900.d have rank \(1\).
Complex multiplication
Each elliptic curve in class 900.d has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).Modular form 900.2.a.d
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.