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SageMath
E = EllipticCurve("cd1")
E.isogeny_class()
Elliptic curves in class 76176.cd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
76176.cd1 | 76176j2 | \([0, 0, 0, -1073019, 427818170]\) | \(15043017316604/243\) | \(2207075890176\) | \([2]\) | \(737280\) | \(1.9140\) | |
76176.cd2 | 76176j1 | \([0, 0, 0, -66999, 6698198]\) | \(-14647977776/59049\) | \(-134079860328192\) | \([2]\) | \(368640\) | \(1.5674\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 76176.cd have rank \(1\).
Complex multiplication
The elliptic curves in class 76176.cd do not have complex multiplication.Modular form 76176.2.a.cd
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.