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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 6760.j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6760.j1 | 6760i3 | \([0, 0, 0, -95147, -11288186]\) | \(9636491538/8125\) | \(80318101760000\) | \([2]\) | \(21504\) | \(1.5956\) | |
6760.j2 | 6760i2 | \([0, 0, 0, -7267, -92274]\) | \(8586756/4225\) | \(20882706457600\) | \([2, 2]\) | \(10752\) | \(1.2490\) | |
6760.j3 | 6760i1 | \([0, 0, 0, -3887, 92274]\) | \(5256144/65\) | \(80318101760\) | \([4]\) | \(5376\) | \(0.90245\) | \(\Gamma_0(N)\)-optimal |
6760.j4 | 6760i4 | \([0, 0, 0, 26533, -707434]\) | \(208974222/142805\) | \(-1411670956533760\) | \([2]\) | \(21504\) | \(1.5956\) |
Rank
sage: E.rank()
The elliptic curves in class 6760.j have rank \(1\).
Complex multiplication
The elliptic curves in class 6760.j do not have complex multiplication.Modular form 6760.2.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 LMFDB 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 LMFDB labels.