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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 465690.g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
465690.g1 | 465690g3 | \([1, 1, 0, -12384473, 7281986577]\) | \(4465136636671380769/2096375976562500\) | \(98625854724618164062500\) | \([2]\) | \(52254720\) | \(3.1064\) | \(\Gamma_0(N)\)-optimal* |
465690.g2 | 465690g1 | \([1, 1, 0, -6341333, -6148716627]\) | \(599437478278595809/33854760000\) | \(1592727010243560000\) | \([2]\) | \(17418240\) | \(2.5571\) | \(\Gamma_0(N)\)-optimal* |
465690.g3 | 465690g2 | \([1, 1, 0, -5980333, -6879164027]\) | \(-502780379797811809/143268096832200\) | \(-6740173834664158168200\) | \([2]\) | \(34836480\) | \(2.9037\) | |
465690.g4 | 465690g4 | \([1, 1, 0, 44021777, 55193455327]\) | \(200541749524551119231/144008551960031250\) | \(-6775009198493946943781250\) | \([2]\) | \(104509440\) | \(3.4530\) |
Rank
sage: E.rank()
The elliptic curves in class 465690.g have rank \(1\).
Complex multiplication
The elliptic curves in class 465690.g do not have complex multiplication.Modular form 465690.2.a.g
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.