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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 7200bg
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
7200.v3 | 7200bg1 | \([0, 0, 0, -225, 0]\) | \(1728\) | \(729000000\) | \([2, 2]\) | \(2048\) | \(0.39007\) | \(\Gamma_0(N)\)-optimal | \(-4\) |
7200.v1 | 7200bg2 | \([0, 0, 0, -2475, -47250]\) | \(287496\) | \(5832000000\) | \([2]\) | \(4096\) | \(0.73664\) | \(-16\) | |
7200.v2 | 7200bg3 | \([0, 0, 0, -2475, 47250]\) | \(287496\) | \(5832000000\) | \([2]\) | \(4096\) | \(0.73664\) | \(-16\) | |
7200.v4 | 7200bg4 | \([0, 0, 0, 900, 0]\) | \(1728\) | \(-46656000000\) | \([2]\) | \(4096\) | \(0.73664\) | \(-4\) |
Rank
sage: E.rank()
The elliptic curves in class 7200bg have rank \(1\).
Complex multiplication
Each elliptic curve in class 7200bg has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-1}) \).Modular form 7200.2.a.bg
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 2 & 2 \\ 2 & 1 & 4 & 4 \\ 2 & 4 & 1 & 4 \\ 2 & 4 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.