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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 15631.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
15631.a1 | 15631c3 | \([1, -1, 1, -2614429, -1626264454]\) | \(16798320881842096017/2132227789307\) | \(250854467184179243\) | \([2]\) | \(258048\) | \(2.3605\) | |
15631.a2 | 15631c4 | \([1, -1, 1, -1037119, 390180018]\) | \(1048626554636928177/48569076788309\) | \(5714103315067765541\) | \([2]\) | \(258048\) | \(2.3605\) | |
15631.a3 | 15631c2 | \([1, -1, 1, -177414, -20758972]\) | \(5249244962308257/1448621666569\) | \(170428890450176281\) | \([2, 2]\) | \(129024\) | \(2.0139\) | |
15631.a4 | 15631c1 | \([1, -1, 1, 28631, -2132504]\) | \(22062729659823/29354283343\) | \(-3453502081020607\) | \([4]\) | \(64512\) | \(1.6673\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 15631.a have rank \(1\).
Complex multiplication
The elliptic curves in class 15631.a do not have complex multiplication.Modular form 15631.2.a.a
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.