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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 388542.n
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
388542.n1 | 388542n4 | \([1, 1, 0, -11885870, -15776129754]\) | \(312196988566716625/25367712678\) | \(15089307101301763638\) | \([2]\) | \(13934592\) | \(2.7252\) | |
388542.n2 | 388542n3 | \([1, 1, 0, -692160, -281796372]\) | \(-61653281712625/21875235228\) | \(-13011900065975152188\) | \([2]\) | \(6967296\) | \(2.3786\) | |
388542.n3 | 388542n2 | \([1, 1, 0, -305300, 32438232]\) | \(5290763640625/2291573592\) | \(1363081414309339032\) | \([2]\) | \(4644864\) | \(2.1759\) | |
388542.n4 | 388542n1 | \([1, 1, 0, 64740, 3797136]\) | \(50447927375/39517632\) | \(-23506009104295872\) | \([2]\) | \(2322432\) | \(1.8293\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 388542.n have rank \(1\).
Complex multiplication
The elliptic curves in class 388542.n do not have complex multiplication.Modular form 388542.2.a.n
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.