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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 3630k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3630.k5 | 3630k1 | \([1, 0, 1, 602, -22024]\) | \(13651919/126720\) | \(-224492209920\) | \([2]\) | \(3840\) | \(0.85932\) | \(\Gamma_0(N)\)-optimal |
| 3630.k4 | 3630k2 | \([1, 0, 1, -9078, -308552]\) | \(46694890801/3920400\) | \(6945227744400\) | \([2, 2]\) | \(7680\) | \(1.2059\) | |
| 3630.k2 | 3630k3 | \([1, 0, 1, -142178, -20646232]\) | \(179415687049201/1443420\) | \(2557106578620\) | \([2]\) | \(15360\) | \(1.5525\) | |
| 3630.k3 | 3630k4 | \([1, 0, 1, -30858, 1730056]\) | \(1834216913521/329422500\) | \(583592053522500\) | \([2, 2]\) | \(15360\) | \(1.5525\) | |
| 3630.k1 | 3630k5 | \([1, 0, 1, -470088, 124011688]\) | \(6484907238722641/283593750\) | \(502403627343750\) | \([2]\) | \(30720\) | \(1.8990\) | |
| 3630.k6 | 3630k6 | \([1, 0, 1, 59892, 10006456]\) | \(13411719834479/32153832150\) | \(-56962475037486150\) | \([2]\) | \(30720\) | \(1.8990\) |
Rank
sage: E.rank()
The elliptic curves in class 3630k have rank \(0\).
Complex multiplication
The elliptic curves in class 3630k do not have complex multiplication.Modular form 3630.2.a.k
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
