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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 2277.b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 2277.b1 | 2277b5 | \([1, -1, 0, -837936, 295442235]\) | \(89254274298475942657/17457\) | \(12726153\) | \([2]\) | \(8192\) | \(1.6644\) | |
| 2277.b2 | 2277b4 | \([1, -1, 0, -52371, 4626072]\) | \(21790813729717297/304746849\) | \(222160452921\) | \([2, 2]\) | \(4096\) | \(1.3178\) | |
| 2277.b3 | 2277b6 | \([1, -1, 0, -50886, 4899609]\) | \(-19989223566735457/2584262514273\) | \(-1883927372905017\) | \([2]\) | \(8192\) | \(1.6644\) | |
| 2277.b4 | 2277b3 | \([1, -1, 0, -12681, -473526]\) | \(309368403125137/44372288367\) | \(32347398219543\) | \([2]\) | \(4096\) | \(1.3178\) | |
| 2277.b5 | 2277b2 | \([1, -1, 0, -3366, 68607]\) | \(5786435182177/627352209\) | \(457339760361\) | \([2, 2]\) | \(2048\) | \(0.97127\) | |
| 2277.b6 | 2277b1 | \([1, -1, 0, 279, 5184]\) | \(3288008303/18259263\) | \(-13311002727\) | \([2]\) | \(1024\) | \(0.62469\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 2277.b have rank \(1\).
Complex multiplication
The elliptic curves in class 2277.b do not have complex multiplication.Modular form 2277.2.a.b
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
