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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 6270q
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6270.o2 | 6270q1 | \([1, 0, 0, -966, 11556]\) | \(-99697252461409/801155520\) | \(-801155520\) | \([2]\) | \(4608\) | \(0.53624\) | \(\Gamma_0(N)\)-optimal |
6270.o1 | 6270q2 | \([1, 0, 0, -15486, 740460]\) | \(410717520667800289/26208600\) | \(26208600\) | \([2]\) | \(9216\) | \(0.88282\) |
Rank
sage: E.rank()
The elliptic curves in class 6270q have rank \(1\).
Complex multiplication
The elliptic curves in class 6270q do not have complex multiplication.Modular form 6270.2.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 Cremona 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 Cremona labels.