Show commands:
SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 51376.z
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 51376.z1 | 51376w2 | \([0, -1, 0, -324198, 71158151]\) | \(-48795070432000/41743\) | \(-3223767809392\) | \([]\) | \(217728\) | \(1.7018\) | |
| 51376.z2 | 51376w1 | \([0, -1, 0, -3098, 143675]\) | \(-42592000/89167\) | \(-6886273249648\) | \([]\) | \(72576\) | \(1.1525\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 51376.z have rank \(1\).
Complex multiplication
The elliptic curves in class 51376.z do not have complex multiplication.Modular form 51376.2.a.z
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
