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SageMath
E = EllipticCurve("de1")
E.isogeny_class()
Elliptic curves in class 162240de
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 162240.ea4 | 162240de1 | \([0, -1, 0, -13745, -2196975]\) | \(-3631696/24375\) | \(-1927634442240000\) | \([2]\) | \(1032192\) | \(1.6163\) | \(\Gamma_0(N)\)-optimal |
| 162240.ea3 | 162240de2 | \([0, -1, 0, -351745, -80004575]\) | \(15214885924/38025\) | \(12028438919577600\) | \([2, 2]\) | \(2064384\) | \(1.9629\) | |
| 162240.ea2 | 162240de3 | \([0, -1, 0, -486945, -12702015]\) | \(20183398562/11567205\) | \(7318102238671011840\) | \([2]\) | \(4128768\) | \(2.3094\) | |
| 162240.ea1 | 162240de4 | \([0, -1, 0, -5624545, -5132401535]\) | \(31103978031362/195\) | \(123368604303360\) | \([2]\) | \(4128768\) | \(2.3094\) |
Rank
sage: E.rank()
The elliptic curves in class 162240de have rank \(1\).
Complex multiplication
The elliptic curves in class 162240de do not have complex multiplication.Modular form 162240.2.a.de
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
