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SageMath
E = EllipticCurve("gn1")
E.isogeny_class()
Elliptic curves in class 62400gn
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
62400.ew2 | 62400gn1 | \([0, 1, 0, -84033, 9840063]\) | \(-16022066761/998400\) | \(-4089446400000000\) | \([2]\) | \(368640\) | \(1.7497\) | \(\Gamma_0(N)\)-optimal |
62400.ew1 | 62400gn2 | \([0, 1, 0, -1364033, 612720063]\) | \(68523370149961/243360\) | \(996802560000000\) | \([2]\) | \(737280\) | \(2.0963\) |
Rank
sage: E.rank()
The elliptic curves in class 62400gn have rank \(1\).
Complex multiplication
The elliptic curves in class 62400gn do not have complex multiplication.Modular form 62400.2.a.gn
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.