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SageMath
E = EllipticCurve("dq1")
E.isogeny_class()
Elliptic curves in class 72450.dq
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 72450.dq1 | 72450dm4 | \([1, -1, 1, -1010480, 391211147]\) | \(10017490085065009/235066440\) | \(2677553668125000\) | \([2]\) | \(1179648\) | \(2.0723\) | |
| 72450.dq2 | 72450dm3 | \([1, -1, 1, -272480, -48996853]\) | \(196416765680689/22365315000\) | \(254754916171875000\) | \([2]\) | \(1179648\) | \(2.0723\) | |
| 72450.dq3 | 72450dm2 | \([1, -1, 1, -65480, 5651147]\) | \(2725812332209/373262400\) | \(4251692025000000\) | \([2, 2]\) | \(589824\) | \(1.7257\) | |
| 72450.dq4 | 72450dm1 | \([1, -1, 1, 6520, 467147]\) | \(2691419471/9891840\) | \(-112674240000000\) | \([2]\) | \(294912\) | \(1.3792\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 72450.dq have rank \(1\).
Complex multiplication
The elliptic curves in class 72450.dq do not have complex multiplication.Modular form 72450.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
