Show commands:
SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 192a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 192.a3 | 192a1 | \([0, -1, 0, -4, -2]\) | \(140608/3\) | \(192\) | \([2]\) | \(8\) | \(-0.77279\) | \(\Gamma_0(N)\)-optimal |
| 192.a2 | 192a2 | \([0, -1, 0, -9, 9]\) | \(21952/9\) | \(36864\) | \([2, 2]\) | \(16\) | \(-0.42622\) | |
| 192.a1 | 192a3 | \([0, -1, 0, -129, 609]\) | \(7301384/3\) | \(98304\) | \([4]\) | \(32\) | \(-0.079647\) | |
| 192.a4 | 192a4 | \([0, -1, 0, 31, 33]\) | \(97336/81\) | \(-2654208\) | \([2]\) | \(32\) | \(-0.079647\) |
Rank
sage: E.rank()
The elliptic curves in class 192a have rank \(1\).
Complex multiplication
The elliptic curves in class 192a do not have complex multiplication.Modular form 192.2.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.
