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SageMath
E = EllipticCurve("dq1")
E.isogeny_class()
Elliptic curves in class 342720dq
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 342720.dq4 | 342720dq1 | \([0, 0, 0, 8999412, 3493518352]\) | \(421792317902132351/271682182840320\) | \(-51919273666961284792320\) | \([2]\) | \(27525120\) | \(3.0476\) | \(\Gamma_0(N)\)-optimal |
| 342720.dq3 | 342720dq2 | \([0, 0, 0, -38186508, 28728548368]\) | \(32224493437735955329/16782725759385600\) | \(3207228838010448091545600\) | \([2, 2]\) | \(55050240\) | \(3.3942\) | |
| 342720.dq1 | 342720dq3 | \([0, 0, 0, -486084108, 4120900181008]\) | \(66464620505913166201729/74880071980801920\) | \(14309804598625461778513920\) | \([2]\) | \(110100480\) | \(3.7407\) | |
| 342720.dq2 | 342720dq4 | \([0, 0, 0, -345263628, -2448401163248]\) | \(23818189767728437646209/232359312482640000\) | \(44404556116746452336640000\) | \([2]\) | \(110100480\) | \(3.7407\) |
Rank
sage: E.rank()
The elliptic curves in class 342720dq have rank \(1\).
Complex multiplication
The elliptic curves in class 342720dq do not have complex multiplication.Modular form 342720.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
