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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 47040bd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 47040.cy4 | 47040bd1 | \([0, -1, 0, 180, 2430]\) | \(85184/405\) | \(-3049462080\) | \([2]\) | \(24576\) | \(0.50218\) | \(\Gamma_0(N)\)-optimal |
| 47040.cy3 | 47040bd2 | \([0, -1, 0, -2025, 31977]\) | \(1906624/225\) | \(108425318400\) | \([2, 2]\) | \(49152\) | \(0.84876\) | |
| 47040.cy2 | 47040bd3 | \([0, -1, 0, -7905, -234975]\) | \(14172488/1875\) | \(7228354560000\) | \([2]\) | \(98304\) | \(1.1953\) | |
| 47040.cy1 | 47040bd4 | \([0, -1, 0, -31425, 2154657]\) | \(890277128/15\) | \(57826836480\) | \([2]\) | \(98304\) | \(1.1953\) |
Rank
sage: E.rank()
The elliptic curves in class 47040bd have rank \(1\).
Complex multiplication
The elliptic curves in class 47040bd do not have complex multiplication.Modular form 47040.2.a.bd
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
