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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 160080j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 160080.cp4 | 160080j1 | \([0, 1, 0, -84000, 9092148]\) | \(16003198512756001/488525390625\) | \(2001000000000000\) | \([2]\) | \(786432\) | \(1.7120\) | \(\Gamma_0(N)\)-optimal |
| 160080.cp2 | 160080j2 | \([0, 1, 0, -1334000, 592592148]\) | \(64096096056024006001/62562515625\) | \(256256064000000\) | \([2, 2]\) | \(1572864\) | \(2.0586\) | |
| 160080.cp1 | 160080j3 | \([0, 1, 0, -21344000, 37947260148]\) | \(262537424941059264096001/250125\) | \(1024512000\) | \([2]\) | \(3145728\) | \(2.4051\) | |
| 160080.cp3 | 160080j4 | \([0, 1, 0, -1324000, 601924148]\) | \(-62665433378363916001/2004003001000125\) | \(-8208396292096512000\) | \([4]\) | \(3145728\) | \(2.4051\) |
Rank
sage: E.rank()
The elliptic curves in class 160080j have rank \(0\).
Complex multiplication
The elliptic curves in class 160080j do not have complex multiplication.Modular form 160080.2.a.j
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.
