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SageMath
E = EllipticCurve("dq1")
E.isogeny_class()
Elliptic curves in class 158400dq
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 158400.hw5 | 158400dq1 | \([0, 0, 0, -157800, 40327000]\) | \(-37256083456/38671875\) | \(-451068750000000000\) | \([2]\) | \(1572864\) | \(2.0828\) | \(\Gamma_0(N)\)-optimal |
| 158400.hw4 | 158400dq2 | \([0, 0, 0, -2970300, 1969702000]\) | \(15529488955216/6125625\) | \(1143188640000000000\) | \([2, 2]\) | \(3145728\) | \(2.4294\) | |
| 158400.hw1 | 158400dq3 | \([0, 0, 0, -47520300, 126086002000]\) | \(15897679904620804/2475\) | \(1847577600000000\) | \([2]\) | \(6291456\) | \(2.7760\) | |
| 158400.hw3 | 158400dq4 | \([0, 0, 0, -3420300, 1333402000]\) | \(5927735656804/2401490025\) | \(1792702697702400000000\) | \([2, 2]\) | \(6291456\) | \(2.7760\) | |
| 158400.hw6 | 158400dq5 | \([0, 0, 0, 11159700, 9702322000]\) | \(102949393183198/86815346805\) | \(-129614618257090560000000\) | \([2]\) | \(12582912\) | \(3.1226\) | |
| 158400.hw2 | 158400dq6 | \([0, 0, 0, -25200300, -47758718000]\) | \(1185450336504002/26043266205\) | \(38882388097935360000000\) | \([2]\) | \(12582912\) | \(3.1226\) |
Rank
sage: E.rank()
The elliptic curves in class 158400dq have rank \(0\).
Complex multiplication
The elliptic curves in class 158400dq do not have complex multiplication.Modular form 158400.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
