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SageMath
E = EllipticCurve("ex1")
E.isogeny_class()
Elliptic curves in class 52800.ex
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
52800.ex1 | 52800db3 | \([0, 1, 0, -128833, -17773537]\) | \(57736239625/255552\) | \(1046740992000000\) | \([2]\) | \(331776\) | \(1.7344\) | |
52800.ex2 | 52800db4 | \([0, 1, 0, -64833, -35373537]\) | \(-7357983625/127552392\) | \(-522454597632000000\) | \([2]\) | \(663552\) | \(2.0809\) | |
52800.ex3 | 52800db1 | \([0, 1, 0, -8833, 298463]\) | \(18609625/1188\) | \(4866048000000\) | \([2]\) | \(110592\) | \(1.1851\) | \(\Gamma_0(N)\)-optimal |
52800.ex4 | 52800db2 | \([0, 1, 0, 7167, 1274463]\) | \(9938375/176418\) | \(-722608128000000\) | \([2]\) | \(221184\) | \(1.5316\) |
Rank
sage: E.rank()
The elliptic curves in class 52800.ex have rank \(1\).
Complex multiplication
The elliptic curves in class 52800.ex do not have complex multiplication.Modular form 52800.2.a.ex
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.