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SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 333795u
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 333795.u6 | 333795u1 | \([1, 0, 0, 630014, 95088851]\) | \(1145725929069119/824683607055\) | \(-19905857468458949295\) | \([2]\) | \(8257536\) | \(2.3915\) | \(\Gamma_0(N)\)-optimal |
| 333795.u4 | 333795u2 | \([1, 0, 0, -2839431, 803549520]\) | \(104887600917094801/49135823993025\) | \(1186019342003496456225\) | \([2, 2]\) | \(16515072\) | \(2.7381\) | |
| 333795.u2 | 333795u3 | \([1, 0, 0, -37887906, 89707511205]\) | \(249190794200766398401/169196996731455\) | \(4084004183198269532895\) | \([2]\) | \(33030144\) | \(3.0847\) | |
| 333795.u3 | 333795u4 | \([1, 0, 0, -23302076, -42745051569]\) | \(57971431973034407521/850187506100625\) | \(20521459591441756880625\) | \([2, 2]\) | \(33030144\) | \(3.0847\) | |
| 333795.u5 | 333795u5 | \([1, 0, 0, -2495521, -116321191360]\) | \(-71205555889646641/242120990499609375\) | \(-5844212114532665762109375\) | \([2]\) | \(66060288\) | \(3.4312\) | |
| 333795.u1 | 333795u6 | \([1, 0, 0, -371510951, -2756197530894]\) | \(234933551390769872069521/51655131348975\) | \(1246829297139947141775\) | \([2]\) | \(66060288\) | \(3.4312\) |
Rank
sage: E.rank()
The elliptic curves in class 333795u have rank \(1\).
Complex multiplication
The elliptic curves in class 333795u do not have complex multiplication.Modular form 333795.2.a.u
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
