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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 50025s
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 50025.t4 | 50025s1 | \([1, 0, 1, -131251, -17823727]\) | \(16003198512756001/488525390625\) | \(7633209228515625\) | \([2]\) | \(294912\) | \(1.8236\) | \(\Gamma_0(N)\)-optimal |
| 50025.t2 | 50025s2 | \([1, 0, 1, -2084376, -1158448727]\) | \(64096096056024006001/62562515625\) | \(977539306640625\) | \([2, 2]\) | \(589824\) | \(2.1701\) | |
| 50025.t3 | 50025s3 | \([1, 0, 1, -2068751, -1176667477]\) | \(-62665433378363916001/2004003001000125\) | \(-31312546890626953125\) | \([4]\) | \(1179648\) | \(2.5167\) | |
| 50025.t1 | 50025s4 | \([1, 0, 1, -33350001, -74132417477]\) | \(262537424941059264096001/250125\) | \(3908203125\) | \([2]\) | \(1179648\) | \(2.5167\) |
Rank
sage: E.rank()
The elliptic curves in class 50025s have rank \(0\).
Complex multiplication
The elliptic curves in class 50025s do not have complex multiplication.Modular form 50025.2.a.s
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.
