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SageMath
E = EllipticCurve("bb1")
E.isogeny_class()
Elliptic curves in class 54150bb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 54150.bj8 | 54150bb1 | \([1, 0, 1, 13349, -1821802]\) | \(357911/2160\) | \(-1587798483750000\) | \([2]\) | \(331776\) | \(1.5987\) | \(\Gamma_0(N)\)-optimal |
| 54150.bj6 | 54150bb2 | \([1, 0, 1, -167151, -23842802]\) | \(702595369/72900\) | \(53588198826562500\) | \([2, 2]\) | \(663552\) | \(1.9452\) | |
| 54150.bj7 | 54150bb3 | \([1, 0, 1, -122026, 53681948]\) | \(-273359449/1536000\) | \(-1129101144000000000\) | \([2]\) | \(995328\) | \(2.1480\) | |
| 54150.bj5 | 54150bb4 | \([1, 0, 1, -618401, 161169698]\) | \(35578826569/5314410\) | \(3906579694456406250\) | \([2]\) | \(1327104\) | \(2.2918\) | |
| 54150.bj4 | 54150bb5 | \([1, 0, 1, -2603901, -1617477302]\) | \(2656166199049/33750\) | \(24809351308593750\) | \([2]\) | \(1327104\) | \(2.2918\) | |
| 54150.bj3 | 54150bb6 | \([1, 0, 1, -3010026, 2005969948]\) | \(4102915888729/9000000\) | \(6615827015625000000\) | \([2, 2]\) | \(1990656\) | \(2.4945\) | |
| 54150.bj1 | 54150bb7 | \([1, 0, 1, -48135026, 128536469948]\) | \(16778985534208729/81000\) | \(59542443140625000\) | \([2]\) | \(3981312\) | \(2.8411\) | |
| 54150.bj2 | 54150bb8 | \([1, 0, 1, -4093026, 433453948]\) | \(10316097499609/5859375000\) | \(4307179046630859375000\) | \([2]\) | \(3981312\) | \(2.8411\) |
Rank
sage: E.rank()
The elliptic curves in class 54150bb have rank \(1\).
Complex multiplication
The elliptic curves in class 54150bb do not have complex multiplication.Modular form 54150.2.a.bb
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
