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SageMath
E = EllipticCurve("gz1")
E.isogeny_class()
Elliptic curves in class 463680gz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 463680.gz3 | 463680gz1 | \([0, 0, 0, -3381636828, 75690009620048]\) | \(358061097267989271289240144/176126855625\) | \(2103647891466240000\) | \([2]\) | \(141557760\) | \(3.7531\) | \(\Gamma_0(N)\)-optimal |
| 463680.gz2 | 463680gz2 | \([0, 0, 0, -3381654828, 75689163555248]\) | \(89516703758060574923008036/1985322833430374025\) | \(94850275447324191242649600\) | \([2, 2]\) | \(283115520\) | \(4.0996\) | |
| 463680.gz4 | 463680gz3 | \([0, 0, 0, -3260932428, 81343269592688]\) | \(-40133926989810174413190818/6689384645060302103835\) | \(-639180656639863715750964756480\) | \([2]\) | \(566231040\) | \(4.4462\) | |
| 463680.gz1 | 463680gz4 | \([0, 0, 0, -3502665228, 69980909370608]\) | \(49737293673675178002921218/6641736806881023047235\) | \(634627834801850391125598535680\) | \([2]\) | \(566231040\) | \(4.4462\) |
Rank
sage: E.rank()
The elliptic curves in class 463680gz have rank \(0\).
Complex multiplication
The elliptic curves in class 463680gz do not have complex multiplication.Modular form 463680.2.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 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.
