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SageMath
E = EllipticCurve("dq1")
E.isogeny_class()
Elliptic curves in class 121680dq
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
121680.bt2 | 121680dq1 | \([0, 0, 0, 96837, -32194838]\) | \(6967871/35100\) | \(-505888383021465600\) | \([2]\) | \(1548288\) | \(2.0793\) | \(\Gamma_0(N)\)-optimal |
121680.bt1 | 121680dq2 | \([0, 0, 0, -1119963, -408672758]\) | \(10779215329/1232010\) | \(17756682244053442560\) | \([2]\) | \(3096576\) | \(2.4259\) |
Rank
sage: E.rank()
The elliptic curves in class 121680dq have rank \(1\).
Complex multiplication
The elliptic curves in class 121680dq do not have complex multiplication.Modular form 121680.2.a.dq
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.