Show commands:
SageMath
E = EllipticCurve("de1")
E.isogeny_class()
Elliptic curves in class 159936de
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 159936.bk4 | 159936de1 | \([0, -1, 0, -138502289, 633259411569]\) | \(-152435594466395827792/1646846627220711\) | \(-3174397687331052395544576\) | \([2]\) | \(26542080\) | \(3.5174\) | \(\Gamma_0(N)\)-optimal |
| 159936.bk3 | 159936de2 | \([0, -1, 0, -2221751009, 40308731283969]\) | \(157304700372188331121828/18069292138401\) | \(139318666906221887422464\) | \([2, 2]\) | \(53084160\) | \(3.8640\) | |
| 159936.bk1 | 159936de3 | \([0, -1, 0, -35548015169, 2579723403506145]\) | \(322159999717985454060440834/4250799\) | \(65549402315292672\) | \([2]\) | \(106168320\) | \(4.2105\) | |
| 159936.bk2 | 159936de4 | \([0, -1, 0, -2227466369, 40090926915873]\) | \(79260902459030376659234/842751810121431609\) | \(12995645631230942559239012352\) | \([2]\) | \(106168320\) | \(4.2105\) |
Rank
sage: E.rank()
The elliptic curves in class 159936de have rank \(1\).
Complex multiplication
The elliptic curves in class 159936de do not have complex multiplication.Modular form 159936.2.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.
