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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 164730d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 164730.dj3 | 164730d1 | \([1, 0, 0, -8965, 324017]\) | \(3301293169/22800\) | \(550336573200\) | \([2]\) | \(327680\) | \(1.0864\) | \(\Gamma_0(N)\)-optimal |
| 164730.dj2 | 164730d2 | \([1, 0, 0, -14745, -146475]\) | \(14688124849/8122500\) | \(196057404202500\) | \([2, 2]\) | \(655360\) | \(1.4330\) | |
| 164730.dj4 | 164730d3 | \([1, 0, 0, 57505, -1143525]\) | \(871257511151/527800050\) | \(-12739810125078450\) | \([2]\) | \(1310720\) | \(1.7795\) | |
| 164730.dj1 | 164730d4 | \([1, 0, 0, -179475, -29237793]\) | \(26487576322129/44531250\) | \(1074876119531250\) | \([2]\) | \(1310720\) | \(1.7795\) |
Rank
sage: E.rank()
The elliptic curves in class 164730d have rank \(0\).
Complex multiplication
The elliptic curves in class 164730d do not have complex multiplication.Modular form 164730.2.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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
