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SageMath
E = EllipticCurve("dd1")
E.isogeny_class()
Elliptic curves in class 47040dd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
47040.gm4 | 47040dd1 | \([0, 1, 0, 180, -2430]\) | \(85184/405\) | \(-3049462080\) | \([2]\) | \(24576\) | \(0.50218\) | \(\Gamma_0(N)\)-optimal |
47040.gm3 | 47040dd2 | \([0, 1, 0, -2025, -31977]\) | \(1906624/225\) | \(108425318400\) | \([2, 2]\) | \(49152\) | \(0.84876\) | |
47040.gm2 | 47040dd3 | \([0, 1, 0, -7905, 234975]\) | \(14172488/1875\) | \(7228354560000\) | \([2]\) | \(98304\) | \(1.1953\) | |
47040.gm1 | 47040dd4 | \([0, 1, 0, -31425, -2154657]\) | \(890277128/15\) | \(57826836480\) | \([2]\) | \(98304\) | \(1.1953\) |
Rank
sage: E.rank()
The elliptic curves in class 47040dd have rank \(0\).
Complex multiplication
The elliptic curves in class 47040dd do not have complex multiplication.Modular form 47040.2.a.dd
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.