Show commands:
SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 152971.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 152971.e1 | 152971e3 | \([0, -1, 1, -197237, -83030645]\) | \(-178643795968/524596891\) | \(-2491889916754514731\) | \([]\) | \(2488320\) | \(2.2169\) | |
| 152971.e2 | 152971e1 | \([0, -1, 1, -12327, 531865]\) | \(-43614208/91\) | \(-432259485931\) | \([]\) | \(276480\) | \(1.1183\) | \(\Gamma_0(N)\)-optimal |
| 152971.e3 | 152971e2 | \([0, -1, 1, 21293, 2611262]\) | \(224755712/753571\) | \(-3579540802994611\) | \([]\) | \(829440\) | \(1.6676\) |
Rank
sage: E.rank()
The elliptic curves in class 152971.e have rank \(1\).
Complex multiplication
The elliptic curves in class 152971.e do not have complex multiplication.Modular form 152971.2.a.e
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.
\(\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
