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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 27744.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 27744.d1 | 27744f4 | \([0, -1, 0, -9344, 350664]\) | \(7301384/3\) | \(37075305984\) | \([2]\) | \(36864\) | \(0.99039\) | |
| 27744.d2 | 27744f3 | \([0, -1, 0, -5009, -132255]\) | \(140608/3\) | \(296602447872\) | \([2]\) | \(36864\) | \(0.99039\) | |
| 27744.d3 | 27744f1 | \([0, -1, 0, -674, 3864]\) | \(21952/9\) | \(13903239744\) | \([2, 2]\) | \(18432\) | \(0.64381\) | \(\Gamma_0(N)\)-optimal |
| 27744.d4 | 27744f2 | \([0, -1, 0, 2216, 25828]\) | \(97336/81\) | \(-1001033261568\) | \([2]\) | \(36864\) | \(0.99039\) |
Rank
sage: E.rank()
The elliptic curves in class 27744.d have rank \(1\).
Complex multiplication
The elliptic curves in class 27744.d do not have complex multiplication.Modular form 27744.2.a.d
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.
