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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 42978.r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
42978.r1 | 42978v1 | \([1, 0, 0, -803575, 284546441]\) | \(-57385609248046791394801/1784804944651399008\) | \(-1784804944651399008\) | \([5]\) | \(1800000\) | \(2.2789\) | \(\Gamma_0(N)\)-optimal |
42978.r2 | 42978v2 | \([1, 0, 0, 3910355, -13981592329]\) | \(6612600178387895114470319/88280159906925363790998\) | \(-88280159906925363790998\) | \([]\) | \(9000000\) | \(3.0837\) |
Rank
sage: E.rank()
The elliptic curves in class 42978.r have rank \(2\).
Complex multiplication
The elliptic curves in class 42978.r do not have complex multiplication.Modular form 42978.2.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 LMFDB numbering.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.