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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 11774.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
11774.a1 | 11774d2 | \([1, 1, 0, -4503543, 3676698901]\) | \(-414183515883649725221/50176\) | \(-1223742464\) | \([]\) | \(123200\) | \(2.0783\) | |
11774.a2 | 11774d1 | \([1, 1, 0, -6078, 309016]\) | \(-1018411856981/1129900996\) | \(-27557155391444\) | \([]\) | \(24640\) | \(1.2735\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 11774.a have rank \(1\).
Complex multiplication
The elliptic curves in class 11774.a do not have complex multiplication.Modular form 11774.2.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.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.