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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 4650.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4650.d1 | 4650f4 | \([1, 1, 0, -15886775, -24379212375]\) | \(28379906689597370652529/1357352437500\) | \(21208631835937500\) | \([2]\) | \(207360\) | \(2.6100\) | |
| 4650.d2 | 4650f3 | \([1, 1, 0, -991275, -382561875]\) | \(-6894246873502147249/47925198774000\) | \(-748831230843750000\) | \([2]\) | \(103680\) | \(2.2635\) | |
| 4650.d3 | 4650f2 | \([1, 1, 0, -213275, -27331875]\) | \(68663623745397169/19216056254400\) | \(300250878975000000\) | \([2]\) | \(69120\) | \(2.0607\) | |
| 4650.d4 | 4650f1 | \([1, 1, 0, 34725, -2779875]\) | \(296354077829711/387386634240\) | \(-6052916160000000\) | \([2]\) | \(34560\) | \(1.7142\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 4650.d have rank \(0\).
Complex multiplication
The elliptic curves in class 4650.d do not have complex multiplication.Modular form 4650.2.a.d
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
