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SageMath
E = EllipticCurve("gg1")
E.isogeny_class()
Elliptic curves in class 162240gg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 162240.fj3 | 162240gg1 | \([0, 1, 0, -146241, 20830239]\) | \(273359449/9360\) | \(11843386013122560\) | \([2]\) | \(1032192\) | \(1.8561\) | \(\Gamma_0(N)\)-optimal |
| 162240.fj2 | 162240gg2 | \([0, 1, 0, -362561, -55357665]\) | \(4165509529/1368900\) | \(1732095204419174400\) | \([2, 2]\) | \(2064384\) | \(2.2027\) | |
| 162240.fj4 | 162240gg3 | \([0, 1, 0, 1043519, -379037281]\) | \(99317171591/106616250\) | \(-134903568805724160000\) | \([2]\) | \(4128768\) | \(2.5492\) | |
| 162240.fj1 | 162240gg4 | \([0, 1, 0, -5229761, -4604242785]\) | \(12501706118329/2570490\) | \(3252489883853783040\) | \([2]\) | \(4128768\) | \(2.5492\) |
Rank
sage: E.rank()
The elliptic curves in class 162240gg have rank \(0\).
Complex multiplication
The elliptic curves in class 162240gg do not have complex multiplication.Modular form 162240.2.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.
