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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 257754r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
257754.r2 | 257754r1 | \([1, 1, 0, 49056535834, 54541328363400852]\) | \(2129503377881546170534943/210835998001488447189504\) | \(-1292649473276724888738086705165320704\) | \([]\) | \(5132626560\) | \(5.6086\) | \(\Gamma_0(N)\)-optimal |
257754.r1 | 257754r2 | \([1, 1, 0, -9493146128126, 11260432875231620892]\) | \(-15431857370630972204702226136417/3779323070318626304351304\) | \(-23171280153759405752526111898134522504\) | \([]\) | \(15397879680\) | \(6.1579\) |
Rank
sage: E.rank()
The elliptic curves in class 257754r have rank \(1\).
Complex multiplication
The elliptic curves in class 257754r do not have complex multiplication.Modular form 257754.2.a.r
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.