Show commands:
SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 392a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 392.d4 | 392a1 | \([0, 0, 0, 49, -686]\) | \(432/7\) | \(-210827008\) | \([4]\) | \(96\) | \(0.27778\) | \(\Gamma_0(N)\)-optimal |
| 392.d3 | 392a2 | \([0, 0, 0, -931, -10290]\) | \(740772/49\) | \(5903156224\) | \([2, 2]\) | \(192\) | \(0.62435\) | |
| 392.d1 | 392a3 | \([0, 0, 0, -14651, -682570]\) | \(1443468546/7\) | \(1686616064\) | \([2]\) | \(384\) | \(0.97092\) | |
| 392.d2 | 392a4 | \([0, 0, 0, -2891, 47334]\) | \(11090466/2401\) | \(578509309952\) | \([2]\) | \(384\) | \(0.97092\) |
Rank
sage: E.rank()
The elliptic curves in class 392a have rank \(1\).
Complex multiplication
The elliptic curves in class 392a do not have complex multiplication.Modular form 392.2.a.a
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.
