Show commands:
SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 3870.z
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3870.z1 | 3870y3 | \([1, -1, 1, -7547, -248731]\) | \(65202655558249/512820150\) | \(373845889350\) | \([2]\) | \(6144\) | \(1.0488\) | |
| 3870.z2 | 3870y2 | \([1, -1, 1, -797, 2369]\) | \(76711450249/41602500\) | \(30328222500\) | \([2, 2]\) | \(3072\) | \(0.70222\) | |
| 3870.z3 | 3870y1 | \([1, -1, 1, -617, 6041]\) | \(35578826569/51600\) | \(37616400\) | \([4]\) | \(1536\) | \(0.35565\) | \(\Gamma_0(N)\)-optimal |
| 3870.z4 | 3870y4 | \([1, -1, 1, 3073, 16301]\) | \(4403686064471/2721093750\) | \(-1983677343750\) | \([2]\) | \(6144\) | \(1.0488\) |
Rank
sage: E.rank()
The elliptic curves in class 3870.z have rank \(0\).
Complex multiplication
The elliptic curves in class 3870.z do not have complex multiplication.Modular form 3870.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
