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SageMath
E = EllipticCurve("dq1")
E.isogeny_class()
Elliptic curves in class 463680dq
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 463680.dq4 | 463680dq1 | \([0, 0, 0, -1060428, 418513552]\) | \(690080604747409/3406760000\) | \(651041974517760000\) | \([2]\) | \(11501568\) | \(2.2658\) | \(\Gamma_0(N)\)-optimal |
| 463680.dq3 | 463680dq2 | \([0, 0, 0, -1636428, -86292848]\) | \(2535986675931409/1450751712200\) | \(277242969638515507200\) | \([2]\) | \(23003136\) | \(2.6123\) | |
| 463680.dq2 | 463680dq3 | \([0, 0, 0, -6094668, -5489140592]\) | \(131010595463836369/7704101562500\) | \(1472276736000000000000\) | \([2]\) | \(34504704\) | \(2.8151\) | |
| 463680.dq1 | 463680dq4 | \([0, 0, 0, -96094668, -362573140592]\) | \(513516182162686336369/1944885031250\) | \(371673317449728000000\) | \([2]\) | \(69009408\) | \(3.1617\) |
Rank
sage: E.rank()
The elliptic curves in class 463680dq have rank \(0\).
Complex multiplication
The elliptic curves in class 463680dq do not have complex multiplication.Modular form 463680.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
