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SageMath
E = EllipticCurve("dd1")
E.isogeny_class()
Elliptic curves in class 250470.dd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 250470.dd1 | 250470dd3 | \([1, -1, 1, -120225623, 507422115731]\) | \(148809678420065817601/20700\) | \(26733386958300\) | \([2]\) | \(15728640\) | \(2.9013\) | |
| 250470.dd2 | 250470dd6 | \([1, -1, 1, -28128893, -49179849793]\) | \(1905890658841300321/293666194803750\) | \(379260484167157366083750\) | \([2]\) | \(31457280\) | \(3.2478\) | |
| 250470.dd3 | 250470dd4 | \([1, -1, 1, -7710143, 7494432707]\) | \(39248884582600321/3935264062500\) | \(5082267486275564062500\) | \([2, 2]\) | \(15728640\) | \(2.9013\) | |
| 250470.dd4 | 250470dd2 | \([1, -1, 1, -7514123, 7929832331]\) | \(36330796409313601/428490000\) | \(553381110036810000\) | \([2, 2]\) | \(7864320\) | \(2.5547\) | |
| 250470.dd5 | 250470dd1 | \([1, -1, 1, -457403, 130745387]\) | \(-8194759433281/965779200\) | \(-1247272901926444800\) | \([2]\) | \(3932160\) | \(2.2081\) | \(\Gamma_0(N)\)-optimal |
| 250470.dd6 | 250470dd5 | \([1, -1, 1, 9572287, 36300787031]\) | \(75108181893694559/484313964843750\) | \(-625475972535095214843750\) | \([2]\) | \(31457280\) | \(3.2478\) |
Rank
sage: E.rank()
The elliptic curves in class 250470.dd have rank \(1\).
Complex multiplication
The elliptic curves in class 250470.dd do not have complex multiplication.Modular form 250470.2.a.dd
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
