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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 3630.w
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3630.w1 | 3630w8 | \([1, 0, 0, -645356, 199494120]\) | \(16778985534208729/81000\) | \(143496441000\) | \([2]\) | \(34560\) | \(1.7631\) | |
| 3630.w2 | 3630w7 | \([1, 0, 0, -54876, 668856]\) | \(10316097499609/5859375000\) | \(10380240234375000\) | \([2]\) | \(34560\) | \(1.7631\) | |
| 3630.w3 | 3630w6 | \([1, 0, 0, -40356, 3111120]\) | \(4102915888729/9000000\) | \(15944049000000\) | \([2, 2]\) | \(17280\) | \(1.4165\) | |
| 3630.w4 | 3630w4 | \([1, 0, 0, -34911, -2513565]\) | \(2656166199049/33750\) | \(59790183750\) | \([2]\) | \(11520\) | \(1.2138\) | |
| 3630.w5 | 3630w5 | \([1, 0, 0, -8291, 249591]\) | \(35578826569/5314410\) | \(9414801494010\) | \([2]\) | \(11520\) | \(1.2138\) | |
| 3630.w6 | 3630w2 | \([1, 0, 0, -2241, -37179]\) | \(702595369/72900\) | \(129146796900\) | \([2, 2]\) | \(5760\) | \(0.86723\) | |
| 3630.w7 | 3630w3 | \([1, 0, 0, -1636, 83216]\) | \(-273359449/1536000\) | \(-2721117696000\) | \([2]\) | \(8640\) | \(1.0700\) | |
| 3630.w8 | 3630w1 | \([1, 0, 0, 179, -2815]\) | \(357911/2160\) | \(-3826571760\) | \([2]\) | \(2880\) | \(0.52066\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 3630.w have rank \(0\).
Complex multiplication
The elliptic curves in class 3630.w do not have complex multiplication.Modular form 3630.2.a.w
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 12 & 3 & 6 & 4 & 12 \\ 4 & 1 & 2 & 3 & 12 & 6 & 4 & 12 \\ 2 & 2 & 1 & 6 & 6 & 3 & 2 & 6 \\ 12 & 3 & 6 & 1 & 4 & 2 & 12 & 4 \\ 3 & 12 & 6 & 4 & 1 & 2 & 12 & 4 \\ 6 & 6 & 3 & 2 & 2 & 1 & 6 & 2 \\ 4 & 4 & 2 & 12 & 12 & 6 & 1 & 3 \\ 12 & 12 & 6 & 4 & 4 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
