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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 900b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
900.g4 | 900b1 | \([0, 0, 0, 0, 125]\) | \(0\) | \(-6750000\) | \([2]\) | \(144\) | \(-0.010696\) | \(\Gamma_0(N)\)-optimal | \(-3\) |
900.g2 | 900b2 | \([0, 0, 0, -375, 2750]\) | \(54000\) | \(108000000\) | \([2]\) | \(288\) | \(0.33588\) | \(-12\) | |
900.g3 | 900b3 | \([0, 0, 0, 0, -3375]\) | \(0\) | \(-4920750000\) | \([2]\) | \(432\) | \(0.53861\) | \(-3\) | |
900.g1 | 900b4 | \([0, 0, 0, -3375, -74250]\) | \(54000\) | \(78732000000\) | \([2]\) | \(864\) | \(0.88518\) | \(-12\) |
Rank
sage: E.rank()
The elliptic curves in class 900b have rank \(0\).
Complex multiplication
Each elliptic curve in class 900b has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).Modular form 900.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.