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SageMath
E = EllipticCurve("bz1")
E.isogeny_class()
Elliptic curves in class 281775bz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 281775.bz6 | 281775bz1 | \([1, 0, 1, -794901, 272708323]\) | \(147281603041/5265\) | \(1985692199765625\) | \([2]\) | \(2949120\) | \(2.0244\) | \(\Gamma_0(N)\)-optimal |
| 281775.bz5 | 281775bz2 | \([1, 0, 1, -831026, 246553823]\) | \(168288035761/27720225\) | \(10454669431766015625\) | \([2, 2]\) | \(5898240\) | \(2.3710\) | |
| 281775.bz7 | 281775bz3 | \([1, 0, 1, 1517099, 1387742573]\) | \(1023887723039/2798036865\) | \(-1055278248335643515625\) | \([2]\) | \(11796480\) | \(2.7176\) | |
| 281775.bz4 | 281775bz4 | \([1, 0, 1, -3757151, -2568378427]\) | \(15551989015681/1445900625\) | \(545320720360634765625\) | \([2, 2]\) | \(11796480\) | \(2.7176\) | |
| 281775.bz8 | 281775bz5 | \([1, 0, 1, 4370974, -12159565927]\) | \(24487529386319/183539412225\) | \(-69221800418755953515625\) | \([2]\) | \(23592960\) | \(3.0642\) | |
| 281775.bz2 | 281775bz6 | \([1, 0, 1, -58703276, -173121150427]\) | \(59319456301170001/594140625\) | \(224079848931884765625\) | \([2, 2]\) | \(23592960\) | \(3.0642\) | |
| 281775.bz3 | 281775bz7 | \([1, 0, 1, -57294401, -181825180177]\) | \(-55150149867714721/5950927734375\) | \(-2244389512538909912109375\) | \([2]\) | \(47185920\) | \(3.4107\) | |
| 281775.bz1 | 281775bz8 | \([1, 0, 1, -939250151, -11079574744177]\) | \(242970740812818720001/24375\) | \(9193019443359375\) | \([2]\) | \(47185920\) | \(3.4107\) |
Rank
sage: E.rank()
The elliptic curves in class 281775bz have rank \(0\).
Complex multiplication
The elliptic curves in class 281775bz do not have complex multiplication.Modular form 281775.2.a.bz
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 8 & 8 & 16 & 16 \\ 4 & 2 & 4 & 1 & 2 & 2 & 4 & 4 \\ 8 & 4 & 8 & 2 & 1 & 4 & 8 & 8 \\ 8 & 4 & 8 & 2 & 4 & 1 & 2 & 2 \\ 16 & 8 & 16 & 4 & 8 & 2 & 1 & 4 \\ 16 & 8 & 16 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
