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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 124950z
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 124950.cc6 | 124950z1 | \([1, 1, 0, -98025, -14554875]\) | \(-56667352321/16711680\) | \(-30720506880000000\) | \([2]\) | \(1179648\) | \(1.8781\) | \(\Gamma_0(N)\)-optimal |
| 124950.cc5 | 124950z2 | \([1, 1, 0, -1666025, -828346875]\) | \(278202094583041/16646400\) | \(30600504900000000\) | \([2, 2]\) | \(2359296\) | \(2.2247\) | |
| 124950.cc4 | 124950z3 | \([1, 1, 0, -1764025, -725544875]\) | \(330240275458561/67652010000\) | \(124362364445156250000\) | \([2, 2]\) | \(4718592\) | \(2.5713\) | |
| 124950.cc2 | 124950z4 | \([1, 1, 0, -26656025, -52982476875]\) | \(1139466686381936641/4080\) | \(7500123750000\) | \([2]\) | \(4718592\) | \(2.5713\) | |
| 124950.cc7 | 124950z5 | \([1, 1, 0, 3748475, -4347257375]\) | \(3168685387909439/6278181696900\) | \(-11540965600915439062500\) | \([2]\) | \(9437184\) | \(2.9178\) | |
| 124950.cc3 | 124950z6 | \([1, 1, 0, -8844525, 9477455625]\) | \(41623544884956481/2962701562500\) | \(5446232439477539062500\) | \([2, 2]\) | \(9437184\) | \(2.9178\) | |
| 124950.cc8 | 124950z7 | \([1, 1, 0, 8023725, 41375316375]\) | \(31077313442863199/420227050781250\) | \(-772488942146301269531250\) | \([2]\) | \(18874368\) | \(3.2644\) | |
| 124950.cc1 | 124950z8 | \([1, 1, 0, -139000775, 630713236875]\) | \(161572377633716256481/914742821250\) | \(1681540284019394531250\) | \([2]\) | \(18874368\) | \(3.2644\) |
Rank
sage: E.rank()
The elliptic curves in class 124950z have rank \(1\).
Complex multiplication
The elliptic curves in class 124950z do not have complex multiplication.Modular form 124950.2.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 Cremona numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 8 & 8 \\ 8 & 4 & 2 & 8 & 4 & 1 & 2 & 2 \\ 16 & 8 & 4 & 16 & 8 & 2 & 1 & 4 \\ 16 & 8 & 4 & 16 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
