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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 227430.z
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 227430.z1 | 227430fz4 | \([1, -1, 0, -2851222290, 58600285736300]\) | \(2768241956450868452043/2058557375000\) | \(1906232519359640107125000\) | \([2]\) | \(134369280\) | \(3.9689\) | |
| 227430.z2 | 227430fz3 | \([1, -1, 0, -177035370, 928235782196]\) | \(-662660286993086283/18441985352000\) | \(-17077353600375864809496000\) | \([2]\) | \(67184640\) | \(3.6223\) | |
| 227430.z3 | 227430fz2 | \([1, -1, 0, -42916650, 42585182836]\) | \(6882017790203934867/3366201047283200\) | \(4275879135099141613478400\) | \([2]\) | \(44789760\) | \(3.4196\) | |
| 227430.z4 | 227430fz1 | \([1, -1, 0, 9760470, 5089608820]\) | \(80956273702840173/55667967918080\) | \(-70711612043016854568960\) | \([2]\) | \(22394880\) | \(3.0730\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 227430.z have rank \(1\).
Complex multiplication
The elliptic curves in class 227430.z do not have complex multiplication.Modular form 227430.2.a.z
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
