Show commands:
SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 330.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 330.d1 | 330c5 | \([1, 1, 1, -171085, -27308713]\) | \(553808571467029327441/12529687500\) | \(12529687500\) | \([2]\) | \(1536\) | \(1.4618\) | |
| 330.d2 | 330c4 | \([1, 1, 1, -11825, 488927]\) | \(182864522286982801/463015182960\) | \(463015182960\) | \([4]\) | \(768\) | \(1.1152\) | |
| 330.d3 | 330c3 | \([1, 1, 1, -10705, -429025]\) | \(135670761487282321/643043610000\) | \(643043610000\) | \([2, 2]\) | \(768\) | \(1.1152\) | |
| 330.d4 | 330c6 | \([1, 1, 1, -5205, -862425]\) | \(-15595206456730321/310672490129100\) | \(-310672490129100\) | \([2]\) | \(1536\) | \(1.4618\) | |
| 330.d5 | 330c2 | \([1, 1, 1, -1025, 767]\) | \(119102750067601/68309049600\) | \(68309049600\) | \([2, 4]\) | \(384\) | \(0.76861\) | |
| 330.d6 | 330c1 | \([1, 1, 1, 255, 255]\) | \(1833318007919/1070530560\) | \(-1070530560\) | \([4]\) | \(192\) | \(0.42204\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 330.d have rank \(0\).
Complex multiplication
The elliptic curves in class 330.d do not have complex multiplication.Modular form 330.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 4 & 8 \\ 8 & 1 & 4 & 8 & 2 & 4 \\ 2 & 4 & 1 & 2 & 2 & 4 \\ 4 & 8 & 2 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
