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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 1785b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1785.j4 | 1785b1 | \([1, 1, 0, 32, -77]\) | \(3449795831/5510295\) | \(-5510295\) | \([2]\) | \(384\) | \(-0.021311\) | \(\Gamma_0(N)\)-optimal |
| 1785.j3 | 1785b2 | \([1, 1, 0, -213, -1008]\) | \(1076575468249/258084225\) | \(258084225\) | \([2, 2]\) | \(768\) | \(0.32526\) | |
| 1785.j1 | 1785b3 | \([1, 1, 0, -3188, -70623]\) | \(3585019225176649/316207395\) | \(316207395\) | \([2]\) | \(1536\) | \(0.67184\) | |
| 1785.j2 | 1785b4 | \([1, 1, 0, -1158, 13923]\) | \(171963096231529/9865918125\) | \(9865918125\) | \([2]\) | \(1536\) | \(0.67184\) |
Rank
sage: E.rank()
The elliptic curves in class 1785b have rank \(0\).
Complex multiplication
The elliptic curves in class 1785b do not have complex multiplication.Modular form 1785.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
