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SageMath
E = EllipticCurve("ff1")
E.isogeny_class()
Elliptic curves in class 223440.ff
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 223440.ff1 | 223440cv4 | \([0, 1, 0, -758536, -254481676]\) | \(100162392144121/23457780\) | \(11304076735365120\) | \([2]\) | \(3538944\) | \(2.0708\) | |
| 223440.ff2 | 223440cv3 | \([0, 1, 0, -350856, 77670900]\) | \(9912050027641/311647500\) | \(150179908515840000\) | \([2]\) | \(3538944\) | \(2.0708\) | |
| 223440.ff3 | 223440cv2 | \([0, 1, 0, -52936, -3005836]\) | \(34043726521/11696400\) | \(5636381751705600\) | \([2, 2]\) | \(1769472\) | \(1.7242\) | |
| 223440.ff4 | 223440cv1 | \([0, 1, 0, 9784, -321420]\) | \(214921799/218880\) | \(-105476149739520\) | \([2]\) | \(884736\) | \(1.3776\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 223440.ff have rank \(0\).
Complex multiplication
The elliptic curves in class 223440.ff do not have complex multiplication.Modular form 223440.2.a.ff
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
