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SageMath
E = EllipticCurve("cy1")
E.isogeny_class()
Elliptic curves in class 230640.cy
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 230640.cy1 | 230640cy6 | \([0, 1, 0, -3075520, -2077018732]\) | \(1770025017602/75\) | \(136320565401600\) | \([2]\) | \(3686400\) | \(2.1970\) | |
| 230640.cy2 | 230640cy4 | \([0, 1, 0, -192520, -32395132]\) | \(868327204/5625\) | \(5112021202560000\) | \([2, 2]\) | \(1843200\) | \(1.8504\) | |
| 230640.cy3 | 230640cy5 | \([0, 1, 0, -77200, -70727500]\) | \(-27995042/1171875\) | \(-2130008834400000000\) | \([2]\) | \(3686400\) | \(2.1970\) | |
| 230640.cy4 | 230640cy2 | \([0, 1, 0, -19540, 194300]\) | \(3631696/2025\) | \(460081908230400\) | \([2, 2]\) | \(921600\) | \(1.5038\) | |
| 230640.cy5 | 230640cy1 | \([0, 1, 0, -14735, 682488]\) | \(24918016/45\) | \(639002650320\) | \([2]\) | \(460800\) | \(1.1573\) | \(\Gamma_0(N)\)-optimal |
| 230640.cy6 | 230640cy3 | \([0, 1, 0, 76560, 1616580]\) | \(54607676/32805\) | \(-29813307653329920\) | \([2]\) | \(1843200\) | \(1.8504\) |
Rank
sage: E.rank()
The elliptic curves in class 230640.cy have rank \(0\).
Complex multiplication
The elliptic curves in class 230640.cy do not have complex multiplication.Modular form 230640.2.a.cy
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
