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SageMath
E = EllipticCurve("ck1")
E.isogeny_class()
Elliptic curves in class 277725.ck
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
277725.ck1 | 277725ck1 | \([1, 1, 0, -3042025, 1753567000]\) | \(1345938541921/203765625\) | \(471322272570556640625\) | \([2]\) | \(9732096\) | \(2.6908\) | \(\Gamma_0(N)\)-optimal |
277725.ck2 | 277725ck2 | \([1, 1, 0, 5223600, 9630707625]\) | \(6814692748079/21258460125\) | \(-49172110052741033203125\) | \([2]\) | \(19464192\) | \(3.0374\) |
Rank
sage: E.rank()
The elliptic curves in class 277725.ck have rank \(1\).
Complex multiplication
The elliptic curves in class 277725.ck do not have complex multiplication.Modular form 277725.2.a.ck
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.