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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 330.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 330.e1 | 330b5 | \([1, 0, 0, -3885, -93525]\) | \(6484907238722641/283593750\) | \(283593750\) | \([2]\) | \(256\) | \(0.70009\) | |
| 330.e2 | 330b4 | \([1, 0, 0, -1175, 15405]\) | \(179415687049201/1443420\) | \(1443420\) | \([4]\) | \(128\) | \(0.35352\) | |
| 330.e3 | 330b3 | \([1, 0, 0, -255, -1323]\) | \(1834216913521/329422500\) | \(329422500\) | \([2, 2]\) | \(128\) | \(0.35352\) | |
| 330.e4 | 330b2 | \([1, 0, 0, -75, 225]\) | \(46694890801/3920400\) | \(3920400\) | \([2, 4]\) | \(64\) | \(0.0069415\) | |
| 330.e5 | 330b1 | \([1, 0, 0, 5, 17]\) | \(13651919/126720\) | \(-126720\) | \([4]\) | \(32\) | \(-0.33963\) | \(\Gamma_0(N)\)-optimal |
| 330.e6 | 330b6 | \([1, 0, 0, 495, -7473]\) | \(13411719834479/32153832150\) | \(-32153832150\) | \([2]\) | \(256\) | \(0.70009\) |
Rank
sage: E.rank()
The elliptic curves in class 330.e have rank \(0\).
Complex multiplication
The elliptic curves in class 330.e do not have complex multiplication.Modular form 330.2.a.e
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
