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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 414050n
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
414050.n4 | 414050n1 | \([1, 0, 1, -48344651, -129214090302]\) | \(1408317602329/2153060\) | \(19104008934365554062500\) | \([2]\) | \(55738368\) | \(3.1750\) | \(\Gamma_0(N)\)-optimal* |
414050.n3 | 414050n2 | \([1, 0, 1, -62836401, -45364824802]\) | \(3092354182009/1689383150\) | \(14989824153143257955468750\) | \([2]\) | \(111476736\) | \(3.5216\) | \(\Gamma_0(N)\)-optimal* |
414050.n2 | 414050n3 | \([1, 0, 1, -196367526, 932971147448]\) | \(94376601570889/12235496000\) | \(108565030654228864625000000\) | \([2]\) | \(167215104\) | \(3.7243\) | \(\Gamma_0(N)\)-optimal* |
414050.n1 | 414050n4 | \([1, 0, 1, -3036750526, 64409850431448]\) | \(349046010201856969/7245875000\) | \(64292337759884076171875000\) | \([2]\) | \(334430208\) | \(4.0709\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 414050n have rank \(0\).
Complex multiplication
The elliptic curves in class 414050n do not have complex multiplication.Modular form 414050.2.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 Cremona 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 Cremona labels.