Show commands:
SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 19992.o
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 19992.o1 | 19992u3 | \([0, -1, 0, -8887003792, -322460981936372]\) | \(322159999717985454060440834/4250799\) | \(1024209411176448\) | \([2]\) | \(6635520\) | \(3.8640\) | |
| 19992.o2 | 19992u4 | \([0, -1, 0, -556866592, -5011087431188]\) | \(79260902459030376659234/842751810121431609\) | \(203056962987983477488109568\) | \([2]\) | \(6635520\) | \(3.8640\) | |
| 19992.o3 | 19992u2 | \([0, -1, 0, -555437752, -5038313691620]\) | \(157304700372188331121828/18069292138401\) | \(2176854170409716990976\) | \([2, 2]\) | \(3317760\) | \(3.5174\) | |
| 19992.o4 | 19992u1 | \([0, -1, 0, -34625572, -79140113660]\) | \(-152435594466395827792/1646846627220711\) | \(-49599963864547693680384\) | \([2]\) | \(1658880\) | \(3.1708\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 19992.o have rank \(1\).
Complex multiplication
The elliptic curves in class 19992.o do not have complex multiplication.Modular form 19992.2.a.o
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
