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SageMath
E = EllipticCurve("gz1")
E.isogeny_class()
Elliptic curves in class 84150.gz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 84150.gz1 | 84150eh4 | \([1, -1, 1, -1297611605, -17991085355603]\) | \(785681552361835673854227/2604236800\) | \(800924889600000000\) | \([2]\) | \(23224320\) | \(3.5414\) | |
| 84150.gz2 | 84150eh3 | \([1, -1, 1, -81099605, -281103659603]\) | \(-191808834096148160787/11043434659840\) | \(-3396373818900480000000\) | \([2]\) | \(11612160\) | \(3.1949\) | |
| 84150.gz3 | 84150eh2 | \([1, -1, 1, -16077230, -24490005603]\) | \(1089365384367428097483/16063552169500000\) | \(6776811071507812500000\) | \([2]\) | \(7741440\) | \(2.9921\) | |
| 84150.gz4 | 84150eh1 | \([1, -1, 1, -105230, -1043109603]\) | \(-305460292990923/1114070936704000\) | \(-469998676422000000000\) | \([2]\) | \(3870720\) | \(2.6455\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 84150.gz have rank \(1\).
Complex multiplication
The elliptic curves in class 84150.gz do not have complex multiplication.Modular form 84150.2.a.gz
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
