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SageMath
E = EllipticCurve("jk1")
E.isogeny_class()
Elliptic curves in class 139650jk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 139650.cc3 | 139650jk1 | \([1, 1, 0, -38000, -2850000]\) | \(3301293169/22800\) | \(41912456250000\) | \([2]\) | \(589824\) | \(1.4474\) | \(\Gamma_0(N)\)-optimal |
| 139650.cc2 | 139650jk2 | \([1, 1, 0, -62500, 1241500]\) | \(14688124849/8122500\) | \(14931312539062500\) | \([2, 2]\) | \(1179648\) | \(1.7940\) | |
| 139650.cc1 | 139650jk3 | \([1, 1, 0, -760750, 254706250]\) | \(26487576322129/44531250\) | \(81860266113281250\) | \([2]\) | \(2359296\) | \(2.1406\) | |
| 139650.cc4 | 139650jk4 | \([1, 1, 0, 243750, 10122750]\) | \(871257511151/527800050\) | \(-970236688788281250\) | \([2]\) | \(2359296\) | \(2.1406\) |
Rank
sage: E.rank()
The elliptic curves in class 139650jk have rank \(1\).
Complex multiplication
The elliptic curves in class 139650jk do not have complex multiplication.Modular form 139650.2.a.jk
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.
