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SageMath
E = EllipticCurve("gg1")
E.isogeny_class()
Elliptic curves in class 162624gg
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
162624.hx4 | 162624gg1 | \([0, 1, 0, 26943, 1046175]\) | \(4657463/3696\) | \(-1716437504753664\) | \([2]\) | \(737280\) | \(1.6112\) | \(\Gamma_0(N)\)-optimal |
162624.hx3 | 162624gg2 | \([0, 1, 0, -127937, 8945055]\) | \(498677257/213444\) | \(99124265899524096\) | \([2, 2]\) | \(1474560\) | \(1.9578\) | |
162624.hx1 | 162624gg3 | \([0, 1, 0, -1754177, 893294367]\) | \(1285429208617/614922\) | \(285572289853390848\) | \([2]\) | \(2949120\) | \(2.3043\) | |
162624.hx2 | 162624gg4 | \([0, 1, 0, -979777, -367397857]\) | \(223980311017/4278582\) | \(1986990966440460288\) | \([2]\) | \(2949120\) | \(2.3043\) |
Rank
sage: E.rank()
The elliptic curves in class 162624gg have rank \(1\).
Complex multiplication
The elliptic curves in class 162624gg do not have complex multiplication.Modular form 162624.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.