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SageMath
E = EllipticCurve("gg1")
E.isogeny_class()
Elliptic curves in class 141120gg
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
141120.mg3 | 141120gg1 | \([0, 0, 0, -146412, 18895184]\) | \(416832723/56000\) | \(46631560937472000\) | \([2]\) | \(884736\) | \(1.9259\) | \(\Gamma_0(N)\)-optimal |
141120.mg4 | 141120gg2 | \([0, 0, 0, 229908, 100029776]\) | \(1613964717/6125000\) | \(-5100326977536000000\) | \([2]\) | \(1769472\) | \(2.2725\) | |
141120.mg1 | 141120gg3 | \([0, 0, 0, -2968812, -1966443696]\) | \(4767078987/6860\) | \(4164314970618593280\) | \([2]\) | \(2654208\) | \(2.4752\) | |
141120.mg2 | 141120gg4 | \([0, 0, 0, -2122092, -3111547824]\) | \(-1740992427/5882450\) | \(-3570900087305443737600\) | \([2]\) | \(5308416\) | \(2.8218\) |
Rank
sage: E.rank()
The elliptic curves in class 141120gg have rank \(0\).
Complex multiplication
The elliptic curves in class 141120gg do not have complex multiplication.Modular form 141120.2.a.gg
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.