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SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 42978.u
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
42978.u1 | 42978r1 | \([1, 0, 0, -459769306, 3796241996132]\) | \(-10748395438529140294639078020769/5737242625602477531070464\) | \(-5737242625602477531070464\) | \([7]\) | \(11063808\) | \(3.7003\) | \(\Gamma_0(N)\)-optimal |
42978.u2 | 42978r2 | \([1, 0, 0, 2127889334, -194909741524108]\) | \(1065542619208351347902742829533791/17028260093251190608019507801424\) | \(-17028260093251190608019507801424\) | \([]\) | \(77446656\) | \(4.6733\) |
Rank
sage: E.rank()
The elliptic curves in class 42978.u have rank \(1\).
Complex multiplication
The elliptic curves in class 42978.u do not have complex multiplication.Modular form 42978.2.a.u
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.
\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.