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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 4830.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4830.a1 | 4830a3 | \([1, 1, 0, -842723, 297413193]\) | \(66187969564358252770489/550144842789780\) | \(550144842789780\) | \([2]\) | \(61440\) | \(1.9982\) | |
4830.a2 | 4830a2 | \([1, 1, 0, -53823, 4415733]\) | \(17244079743478944889/1469997007491600\) | \(1469997007491600\) | \([2, 2]\) | \(30720\) | \(1.6517\) | |
4830.a3 | 4830a1 | \([1, 1, 0, -11503, -400283]\) | \(168351140229842809/29855318411520\) | \(29855318411520\) | \([2]\) | \(15360\) | \(1.3051\) | \(\Gamma_0(N)\)-optimal |
4830.a4 | 4830a4 | \([1, 1, 0, 57957, 20489697]\) | \(21529289381199961031/193397385415972500\) | \(-193397385415972500\) | \([2]\) | \(61440\) | \(1.9982\) |
Rank
sage: E.rank()
The elliptic curves in class 4830.a have rank \(1\).
Complex multiplication
The elliptic curves in class 4830.a do not have complex multiplication.Modular form 4830.2.a.a
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.