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SageMath
E = EllipticCurve("ip1")
E.isogeny_class()
Elliptic curves in class 145200.ip
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 145200.ip1 | 145200ie6 | \([0, 1, 0, -9681008, -11597112012]\) | \(1770025017602/75\) | \(4251746400000000\) | \([2]\) | \(3932160\) | \(2.4837\) | |
| 145200.ip2 | 145200ie4 | \([0, 1, 0, -606008, -180762012]\) | \(868327204/5625\) | \(159440490000000000\) | \([2, 2]\) | \(1966080\) | \(2.1371\) | |
| 145200.ip3 | 145200ie5 | \([0, 1, 0, -243008, -394932012]\) | \(-27995042/1171875\) | \(-66433537500000000000\) | \([2]\) | \(3932160\) | \(2.4837\) | |
| 145200.ip4 | 145200ie2 | \([0, 1, 0, -61508, 1100988]\) | \(3631696/2025\) | \(14349644100000000\) | \([2, 2]\) | \(983040\) | \(1.7905\) | |
| 145200.ip5 | 145200ie1 | \([0, 1, 0, -46383, 3823488]\) | \(24918016/45\) | \(19930061250000\) | \([2]\) | \(491520\) | \(1.4439\) | \(\Gamma_0(N)\)-optimal |
| 145200.ip6 | 145200ie3 | \([0, 1, 0, 240992, 8965988]\) | \(54607676/32805\) | \(-929856937680000000\) | \([2]\) | \(1966080\) | \(2.1371\) |
Rank
sage: E.rank()
The elliptic curves in class 145200.ip have rank \(1\).
Complex multiplication
The elliptic curves in class 145200.ip do not have complex multiplication.Modular form 145200.2.a.ip
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.
