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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 6720.s
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6720.s1 | 6720i3 | \([0, -1, 0, -810465, 272909025]\) | \(898353183174324196/29899176238575\) | \(1959472413971251200\) | \([2]\) | \(122880\) | \(2.2828\) | |
6720.s2 | 6720i2 | \([0, -1, 0, -124465, -10957775]\) | \(13015144447800784/4341909875625\) | \(71137851402240000\) | \([2, 2]\) | \(61440\) | \(1.9362\) | |
6720.s3 | 6720i1 | \([0, -1, 0, -111965, -14380275]\) | \(151591373397612544/32558203125\) | \(33339600000000\) | \([2]\) | \(30720\) | \(1.5897\) | \(\Gamma_0(N)\)-optimal |
6720.s4 | 6720i4 | \([0, -1, 0, 361535, -75984575]\) | \(79743193254623804/84085819746075\) | \(-5510648282878771200\) | \([2]\) | \(122880\) | \(2.2828\) |
Rank
sage: E.rank()
The elliptic curves in class 6720.s have rank \(0\).
Complex multiplication
The elliptic curves in class 6720.s do not have complex multiplication.Modular form 6720.2.a.s
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.