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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 184041.q
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
184041.q1 | 184041o2 | \([0, 0, 1, -1349634, 619200832]\) | \(-32768\) | \(-8297013726470605851\) | \([]\) | \(3104640\) | \(2.4077\) | \(-11\) | |
184041.q2 | 184041o1 | \([0, 0, 1, -11154, -465215]\) | \(-32768\) | \(-4683447945891\) | \([]\) | \(282240\) | \(1.2087\) | \(\Gamma_0(N)\)-optimal | \(-11\) |
Rank
sage: E.rank()
The elliptic curves in class 184041.q have rank \(2\).
Complex multiplication
Each elliptic curve in class 184041.q has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-11}) \).Modular form 184041.2.a.q
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.
\(\left(\begin{array}{rr} 1 & 11 \\ 11 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.