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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 43758.n
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
43758.n1 | 43758t3 | \([1, -1, 1, -29656985, 62160880889]\) | \(3957101249824708884951625/772310238681366528\) | \(563014163998716198912\) | \([6]\) | \(3649536\) | \(2.9814\) | |
43758.n2 | 43758t4 | \([1, -1, 1, -26523545, 75806385401]\) | \(-2830680648734534916567625/1766676274677722124288\) | \(-1287907004240059428605952\) | \([6]\) | \(7299072\) | \(3.3279\) | |
43758.n3 | 43758t1 | \([1, -1, 1, -902930, -213616879]\) | \(111675519439697265625/37528570137307392\) | \(27358327630097088768\) | \([2]\) | \(1216512\) | \(2.4321\) | \(\Gamma_0(N)\)-optimal |
43758.n4 | 43758t2 | \([1, -1, 1, 2634430, -1478576815]\) | \(2773679829880629422375/2899504554614368272\) | \(-2113738820313874470288\) | \([2]\) | \(2433024\) | \(2.7786\) |
Rank
sage: E.rank()
The elliptic curves in class 43758.n have rank \(0\).
Complex multiplication
The elliptic curves in class 43758.n do not have complex multiplication.Modular form 43758.2.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.