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SageMath
E = EllipticCurve("gk1")
E.isogeny_class()
Elliptic curves in class 369600.gk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 369600.gk1 | 369600gk3 | \([0, -1, 0, -221568033, 1269499715937]\) | \(2349497892139423119368/8932840308315\) | \(4573614237857280000000\) | \([2]\) | \(75497472\) | \(3.3706\) | |
| 369600.gk2 | 369600gk4 | \([0, -1, 0, -41836033, -80213516063]\) | \(15816313046221571528/3722207994264375\) | \(1905770493063360000000000\) | \([2]\) | \(75497472\) | \(3.3706\) | |
| 369600.gk3 | 369600gk2 | \([0, -1, 0, -14053033, 19221840937]\) | \(4795721641044996544/282532899951225\) | \(18082105596878400000000\) | \([2, 2]\) | \(37748736\) | \(3.0240\) | |
| 369600.gk4 | 369600gk1 | \([0, -1, 0, 653092, 1236250062]\) | \(30806768067763904/678292279285005\) | \(-678292279285005000000\) | \([2]\) | \(18874368\) | \(2.6775\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 369600.gk have rank \(0\).
Complex multiplication
The elliptic curves in class 369600.gk do not have complex multiplication.Modular form 369600.2.a.gk
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
