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SageMath
E = EllipticCurve("er1")
E.isogeny_class()
Elliptic curves in class 229320.er
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 229320.er1 | 229320dj3 | \([0, 0, 0, -89893587, 328050858814]\) | \(914732517663095044/9555\) | \(839163173022720\) | \([2]\) | \(12582912\) | \(2.8915\) | |
| 229320.er2 | 229320dj6 | \([0, 0, 0, -33833667, -72217661474]\) | \(24385137179326562/1284775885575\) | \(225669619835100374169600\) | \([2]\) | \(25165824\) | \(3.2381\) | |
| 229320.er3 | 229320dj4 | \([0, 0, 0, -6050667, 4291163926]\) | \(278944461825124/70849130625\) | \(6222290032566840960000\) | \([2, 2]\) | \(12582912\) | \(2.8915\) | |
| 229320.er4 | 229320dj2 | \([0, 0, 0, -5618487, 5125530634]\) | \(893359210685776/91298025\) | \(2004551029558022400\) | \([2, 2]\) | \(6291456\) | \(2.5450\) | |
| 229320.er5 | 229320dj1 | \([0, 0, 0, -324282, 92859361]\) | \(-2748251600896/1124136195\) | \(-1542604814733593520\) | \([2]\) | \(3145728\) | \(2.1984\) | \(\Gamma_0(N)\)-optimal |
| 229320.er6 | 229320dj5 | \([0, 0, 0, 14817453, 27400520014]\) | \(2048324060764798/3031899609375\) | \(-532550182415378400000000\) | \([2]\) | \(25165824\) | \(3.2381\) |
Rank
sage: E.rank()
The elliptic curves in class 229320.er have rank \(0\).
Complex multiplication
The elliptic curves in class 229320.er do not have complex multiplication.Modular form 229320.2.a.er
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
