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SageMath
E = EllipticCurve("cu1")
E.isogeny_class()
Elliptic curves in class 58800.cu
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 58800.cu1 | 58800fc7 | \([0, -1, 0, -104537008, -411354663488]\) | \(16778985534208729/81000\) | \(609892416000000000\) | \([2]\) | \(3981312\) | \(3.0350\) | |
| 58800.cu2 | 58800fc8 | \([0, -1, 0, -8889008, -1385383488]\) | \(10316097499609/5859375000\) | \(44118375000000000000000\) | \([2]\) | \(3981312\) | \(3.0350\) | |
| 58800.cu3 | 58800fc6 | \([0, -1, 0, -6537008, -6418663488]\) | \(4102915888729/9000000\) | \(67765824000000000000\) | \([2, 2]\) | \(1990656\) | \(2.6884\) | |
| 58800.cu4 | 58800fc5 | \([0, -1, 0, -5655008, 5177872512]\) | \(2656166199049/33750\) | \(254121840000000000\) | \([2]\) | \(1327104\) | \(2.4857\) | |
| 58800.cu5 | 58800fc4 | \([0, -1, 0, -1343008, -515535488]\) | \(35578826569/5314410\) | \(40015041413760000000\) | \([2]\) | \(1327104\) | \(2.4857\) | |
| 58800.cu6 | 58800fc2 | \([0, -1, 0, -363008, 76384512]\) | \(702595369/72900\) | \(548903174400000000\) | \([2, 2]\) | \(663552\) | \(2.1391\) | |
| 58800.cu7 | 58800fc3 | \([0, -1, 0, -265008, -171751488]\) | \(-273359449/1536000\) | \(-11565367296000000000\) | \([2]\) | \(995328\) | \(2.3418\) | |
| 58800.cu8 | 58800fc1 | \([0, -1, 0, 28992, 5824512]\) | \(357911/2160\) | \(-16263797760000000\) | \([2]\) | \(331776\) | \(1.7925\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 58800.cu have rank \(1\).
Complex multiplication
The elliptic curves in class 58800.cu do not have complex multiplication.Modular form 58800.2.a.cu
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 12 & 3 & 6 & 4 & 12 \\ 4 & 1 & 2 & 3 & 12 & 6 & 4 & 12 \\ 2 & 2 & 1 & 6 & 6 & 3 & 2 & 6 \\ 12 & 3 & 6 & 1 & 4 & 2 & 12 & 4 \\ 3 & 12 & 6 & 4 & 1 & 2 & 12 & 4 \\ 6 & 6 & 3 & 2 & 2 & 1 & 6 & 2 \\ 4 & 4 & 2 & 12 & 12 & 6 & 1 & 3 \\ 12 & 12 & 6 & 4 & 4 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
