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SageMath
E = EllipticCurve("db1")
E.isogeny_class()
Elliptic curves in class 209814db
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 209814.t5 | 209814db1 | \([1, 1, 0, -1189674, -463696812]\) | \(4354703137/352512\) | \(15073827630121675008\) | \([2]\) | \(5898240\) | \(2.4229\) | \(\Gamma_0(N)\)-optimal |
| 209814.t4 | 209814db2 | \([1, 1, 0, -3987194, 2525733060]\) | \(163936758817/30338064\) | \(1297291290417346655376\) | \([2, 2]\) | \(11796480\) | \(2.7695\) | |
| 209814.t2 | 209814db3 | \([1, 1, 0, -60636974, 181708987200]\) | \(576615941610337/27060804\) | \(1157151799168559208036\) | \([2, 2]\) | \(23592960\) | \(3.1161\) | |
| 209814.t6 | 209814db4 | \([1, 1, 0, 7902266, 14721941128]\) | \(1276229915423/2927177028\) | \(-125169531712179579498852\) | \([2]\) | \(23592960\) | \(3.1161\) | |
| 209814.t1 | 209814db5 | \([1, 1, 0, -970180664, 11630863048182]\) | \(2361739090258884097/5202\) | \(222443636902837218\) | \([2]\) | \(47185920\) | \(3.4626\) | |
| 209814.t3 | 209814db6 | \([1, 1, 0, -57489764, 201416186778]\) | \(-491411892194497/125563633938\) | \(-5369248634353179645243042\) | \([2]\) | \(47185920\) | \(3.4626\) |
Rank
sage: E.rank()
The elliptic curves in class 209814db have rank \(0\).
Complex multiplication
The elliptic curves in class 209814db do not have complex multiplication.Modular form 209814.2.a.db
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
