Show commands:
SageMath
E = EllipticCurve("ec1")
E.isogeny_class()
Elliptic curves in class 212160.ec
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 212160.ec1 | 212160ff4 | \([0, 1, 0, -473441, -124757505]\) | \(44769506062996441/323730468750\) | \(84864000000000000\) | \([2]\) | \(4128768\) | \(2.0800\) | |
| 212160.ec2 | 212160ff2 | \([0, 1, 0, -49121, 926079]\) | \(50002789171321/27473062500\) | \(7201898496000000\) | \([2, 2]\) | \(2064384\) | \(1.7334\) | |
| 212160.ec3 | 212160ff1 | \([0, 1, 0, -37601, 2790015]\) | \(22428153804601/35802000\) | \(9385279488000\) | \([2]\) | \(1032192\) | \(1.3868\) | \(\Gamma_0(N)\)-optimal |
| 212160.ec4 | 212160ff3 | \([0, 1, 0, 190879, 7502079]\) | \(2933972022568679/1789082460750\) | \(-468997232590848000\) | \([2]\) | \(4128768\) | \(2.0800\) |
Rank
sage: E.rank()
The elliptic curves in class 212160.ec have rank \(1\).
Complex multiplication
The elliptic curves in class 212160.ec do not have complex multiplication.Modular form 212160.2.a.ec
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
