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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 130536.bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 130536.bc1 | 130536h4 | \([0, 0, 0, -3133011, -2134474594]\) | \(38725206845188/333\) | \(29245561132032\) | \([2]\) | \(1572864\) | \(2.1693\) | |
| 130536.bc2 | 130536h2 | \([0, 0, 0, -195951, -33301870]\) | \(37897488592/110889\) | \(2434692964241664\) | \([2, 2]\) | \(786432\) | \(1.8227\) | |
| 130536.bc3 | 130536h3 | \([0, 0, 0, -116571, -60529210]\) | \(-1994709028/16867449\) | \(-1481375408020816896\) | \([2]\) | \(1572864\) | \(2.1693\) | |
| 130536.bc4 | 130536h1 | \([0, 0, 0, -17346, -45619]\) | \(420616192/242757\) | \(333125219769552\) | \([2]\) | \(393216\) | \(1.4762\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 130536.bc have rank \(1\).
Complex multiplication
The elliptic curves in class 130536.bc do not have complex multiplication.Modular form 130536.2.a.bc
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
