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SageMath
E = EllipticCurve("cl1")
E.isogeny_class()
Elliptic curves in class 151725cl
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 151725.co5 | 151725cl1 | \([1, 0, 1, 252724, 121055573]\) | \(4733169839/19518975\) | \(-7361571966746484375\) | \([2]\) | \(3538944\) | \(2.3029\) | \(\Gamma_0(N)\)-optimal |
| 151725.co4 | 151725cl2 | \([1, 0, 1, -2673401, 1484629823]\) | \(5602762882081/716900625\) | \(270378723470009765625\) | \([2, 2]\) | \(7077888\) | \(2.6494\) | |
| 151725.co2 | 151725cl3 | \([1, 0, 1, -41363276, 102387823823]\) | \(20751759537944401/418359375\) | \(157784035638427734375\) | \([2]\) | \(14155776\) | \(2.9960\) | |
| 151725.co3 | 151725cl4 | \([1, 0, 1, -10801526, -12138107677]\) | \(369543396484081/45120132225\) | \(17017036013594703515625\) | \([2, 2]\) | \(14155776\) | \(2.9960\) | |
| 151725.co6 | 151725cl5 | \([1, 0, 1, 15750349, -62427358927]\) | \(1145725929069119/5127181719135\) | \(-1933714101893120043984375\) | \([2]\) | \(28311552\) | \(3.3426\) | |
| 151725.co1 | 151725cl6 | \([1, 0, 1, -167403401, -833671543927]\) | \(1375634265228629281/24990412335\) | \(9425121907414275234375\) | \([2]\) | \(28311552\) | \(3.3426\) |
Rank
sage: E.rank()
The elliptic curves in class 151725cl have rank \(2\).
Complex multiplication
The elliptic curves in class 151725cl do not have complex multiplication.Modular form 151725.2.a.cl
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.
