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SageMath
E = EllipticCurve("l1")
E.isogeny_class()
Elliptic curves in class 166410.l
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 166410.l1 | 166410ch8 | \([1, -1, 0, -88755120, -321816489800]\) | \(16778985534208729/81000\) | \(373270166680401000\) | \([2]\) | \(15482880\) | \(2.9941\) | |
| 166410.l2 | 166410ch7 | \([1, -1, 0, -7547040, -1084245944]\) | \(10316097499609/5859375000\) | \(27001603492505859375000\) | \([2]\) | \(15482880\) | \(2.9941\) | |
| 166410.l3 | 166410ch6 | \([1, -1, 0, -5550120, -5021772800]\) | \(4102915888729/9000000\) | \(41474462964489000000\) | \([2, 2]\) | \(7741440\) | \(2.6475\) | |
| 166410.l4 | 166410ch4 | \([1, -1, 0, -4801275, 4050484375]\) | \(2656166199049/33750\) | \(155529236116833750\) | \([2]\) | \(5160960\) | \(2.4448\) | |
| 166410.l5 | 166410ch5 | \([1, -1, 0, -1140255, -403379429]\) | \(35578826569/5314410\) | \(24490255635901109610\) | \([2]\) | \(5160960\) | \(2.4448\) | |
| 166410.l6 | 166410ch2 | \([1, -1, 0, -308205, 59739601]\) | \(702595369/72900\) | \(335943150012360900\) | \([2, 2]\) | \(2580480\) | \(2.0982\) | |
| 166410.l7 | 166410ch3 | \([1, -1, 0, -225000, -134377664]\) | \(-273359449/1536000\) | \(-7078308345939456000\) | \([2]\) | \(3870720\) | \(2.3009\) | |
| 166410.l8 | 166410ch1 | \([1, -1, 0, 24615, 4558045]\) | \(357911/2160\) | \(-9953871111477360\) | \([2]\) | \(1290240\) | \(1.7516\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 166410.l have rank \(1\).
Complex multiplication
The elliptic curves in class 166410.l do not have complex multiplication.Modular form 166410.2.a.l
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.
