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SageMath
E = EllipticCurve("df1")
E.isogeny_class()
Elliptic curves in class 29040.df
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 29040.df1 | 29040dg8 | \([0, 1, 0, -4181800, -3292887820]\) | \(1114544804970241/405\) | \(2938807111680\) | \([2]\) | \(327680\) | \(2.1830\) | |
| 29040.df2 | 29040dg6 | \([0, 1, 0, -261400, -51501100]\) | \(272223782641/164025\) | \(1190216880230400\) | \([2, 2]\) | \(163840\) | \(1.8364\) | |
| 29040.df3 | 29040dg7 | \([0, 1, 0, -213000, -71112780]\) | \(-147281603041/215233605\) | \(-1561802590238330880\) | \([2]\) | \(327680\) | \(2.1830\) | |
| 29040.df4 | 29040dg4 | \([0, 1, 0, -154920, 23418228]\) | \(56667352321/15\) | \(108844707840\) | \([2]\) | \(81920\) | \(1.4898\) | |
| 29040.df5 | 29040dg3 | \([0, 1, 0, -19400, -487500]\) | \(111284641/50625\) | \(367350888960000\) | \([2, 2]\) | \(81920\) | \(1.4898\) | |
| 29040.df6 | 29040dg2 | \([0, 1, 0, -9720, 360468]\) | \(13997521/225\) | \(1632670617600\) | \([2, 2]\) | \(40960\) | \(1.1432\) | |
| 29040.df7 | 29040dg1 | \([0, 1, 0, -40, 15860]\) | \(-1/15\) | \(-108844707840\) | \([2]\) | \(20480\) | \(0.79667\) | \(\Gamma_0(N)\)-optimal |
| 29040.df8 | 29040dg5 | \([0, 1, 0, 67720, -3588972]\) | \(4733169839/3515625\) | \(-25510478400000000\) | \([2]\) | \(163840\) | \(1.8364\) |
Rank
sage: E.rank()
The elliptic curves in class 29040.df have rank \(1\).
Complex multiplication
The elliptic curves in class 29040.df do not have complex multiplication.Modular form 29040.2.a.df
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
