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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 759b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 759.b6 | 759b1 | \([1, 0, 0, 31, -192]\) | \(3288008303/18259263\) | \(-18259263\) | \([4]\) | \(128\) | \(0.075387\) | \(\Gamma_0(N)\)-optimal |
| 759.b5 | 759b2 | \([1, 0, 0, -374, -2541]\) | \(5786435182177/627352209\) | \(627352209\) | \([2, 4]\) | \(256\) | \(0.42196\) | |
| 759.b2 | 759b3 | \([1, 0, 0, -5819, -171336]\) | \(21790813729717297/304746849\) | \(304746849\) | \([2, 2]\) | \(512\) | \(0.76853\) | |
| 759.b4 | 759b4 | \([1, 0, 0, -1409, 17538]\) | \(309368403125137/44372288367\) | \(44372288367\) | \([4]\) | \(512\) | \(0.76853\) | |
| 759.b1 | 759b5 | \([1, 0, 0, -93104, -10942305]\) | \(89254274298475942657/17457\) | \(17457\) | \([2]\) | \(1024\) | \(1.1151\) | |
| 759.b3 | 759b6 | \([1, 0, 0, -5654, -181467]\) | \(-19989223566735457/2584262514273\) | \(-2584262514273\) | \([2]\) | \(1024\) | \(1.1151\) |
Rank
sage: E.rank()
The elliptic curves in class 759b have rank \(1\).
Complex multiplication
The elliptic curves in class 759b do not have complex multiplication.Modular form 759.2.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 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.
